Souffle has said it all really
I use the Chinchilla bathing sand in the potty and I empty that about every 5 days. I don't do a cage clean for two months or more! When there is plenty of substrate and they use a litter tray then really they are quite clean little things.
During that time you can clean the odd thing "as and when" eg if the wheel gets pee'd in (which it usually does) you could give that a wipe out now and then but again I wouldn't keep washing it 100% clean too often or it just makes them pee on something more to re-scent mark it. I only do it if it starts to smell! Our last hammy wouldn't use his wheel if I gave it a thorough clean and it took tempting him in there with a treat or something sometimes.
They are generally very clean little things and I've found them even to clean out their own nest sometimes if they think it needs doing - ie a pile of old substrate and nesting material outside the door of the house and a big hollow in the substrate where they have carted some more inside the house!
The exception is if the nest or hoard gets pee'd on. That sometimes happens with baby hamsters who aren't quite toilet trained. Then you need to remove most of the pee'd on nest (but try and leave a bit of the old dry nest behind even if it's slightly whiffy because they do get stressed about the nest and hoard). And also remove any pee'd on hoarded food - but again try and leave some of the dry hoard behind and also replace what you've removed with new food in exactly the same place.
It can become an itch scratch cycle with the hoard. They are very precious about it and if someone removes it during cleaning they can start peeing on the hoard to deter "people or things" from stealing it. Then because it's pee'd on it needs removing and replacing - and so on. You can break the cycle eventually but it's best not to get into it if possible.
You will get to know your hammy's individual habits. Both our Syrians have only ever hoarded dry hamster mix. Any fresh food they eat straight away. In which case it's fine to leave it - but you may need to have a peek now and then to check there is nothing fresh going off in there - I'd leave that until he's settled in a bit. One way round that I had for a while was to only feed fresh food when our hammy was out of the cage. He'd eat a bit and leave the rest. If he pouched it I'd try and encourage him to unpouch it again before he went back in the cage (he usually did if he had been out for a short while - in a little hide or something). If he didn't then I had to check the nest the next day.
Anyway - trial and error - start by giving a bit of fresh veg every 3 days - he should sit and eat it there and then and not pouch it. Then every other day after about a week, and then by two weeks every day (lets their stomachs adjust to veg).
I have a 100cm cage and went four to six months before needing to do a substrate clean last time. It still wasn't that bad as he uses the litter tray.
So partial cleans usually mean - do wheel as and when, toys and when (they rarely need a lot of cleaning) and do the substrate at a different time. ie don't clean everything at the same time or nothing smells familiar to them and they get quite stressed. I tend to leave it a long time, replace the substrate and then spread some of the old clean substrate on top so it still smells familiar. Other options are keep the clean half and mix it in with new or keep two thirds of the old substrate and mix it in with the new - depending how long you've left it. And I then hide treats and add a new chew after a substrate part change to distract our hammy from feeling anxious about the change.
They seem to accept you changing the litter tray fine and it helps if they see you do it so they get used to it.
They tend to choose a pee place - often a corner of the cage. If that is a different place to where you've put the litter tray then move the litter tray to the chosen place - they will only use the litter tray usually if it's in the chosen place!
I use Fitch bedding and also put out a pile of torn up strips of plain white toilet paper and the toilet paper strips are taken for the nest - although sometimes they use some of the fitch as well. Our hammy has just taken loads to build up a big nest for the winter.
Houses - they tend to use one house to nest in but having a few hidey places is good as they like to have places to go and things to do and feel secure. They also like sitting under or on a platform. Hidey places and tunnels can also fill up the open space a bit so they don't feel too exposed from above. Again depends on the personality of the hamster - some aren't bothered, some prefer to be "under" something when moving around the cage and protected from above.
Also think a house is best open underneath and sat on top of the substrate. It means they can have normal behaviours like burying their hoard under their nest (they like to eat in bed
). And can burrow down a bit when it's cold. Sometimes they won't use a house. Our last hamster ignored a little house I put in - presumably it wasn't big enough for his nest or wasn't dark enough inside. When I replaced it with a larger one he moved straight in. With a large house though I've found they then tend to move the pee place inside a corner of the house - so i then had to move the litter tray inside the house as well.
I've gone with - large house - litter tray inside. But medium sized house is probably easiest and the litter tray in the cage in their chosen place. They naturally keep their toilet area separate from their nest usually. I think the main thing wiht the house is they like to feel safe and retreated inside it - the darker it is inside the better - I used to put a bendy bridge over the entrance to make a tunnel and that always seemed to tempt a hammy inside to use it.
Your hammy will check out the cage and decide where is the best place to build a nest. He'll probably use one of the houses. I wouldn't use a house with a solid base as it could need cleaning out too much and cause issues. If he doesn't think either of the houses are ok he may well just build a big pile of substrate under a shelf.
Not sure which houses you've got but a lot of them are too small for syrians. Main thing is check the size of any entrance holes or windows - at least 6cm ideally - anything smaller than 5cm and they can get stuck (eg small round windows). Again depending on the size of the hamster but 6cm upwards is the best entrance size.
A cardboard shoebox can also make a good nesting box if you cut the base out and use the lid as a lift off roof and cut a hole in for a door - then you can lift the top to check inside (I have a lift off roof house and take the litter tray out that way). But it's not necessary really. If there's one of the houses you think would be easier for checking on then maybe try putting a bendy bridge over the door - the dark tunnel entrance tempts them.
Settling in - usual advice is leave them 2 to 3 days to acclimatise to the cage and scent mark. Maybe see how he is after the first day if he's already hand tame, but he may still be a bit stressed by everything being new. Next stage is usually offering them a treat through the bars and then on the palm of your hand. I've always had a problem with the last bit as to do that you have to have your hand in the cage usually and some hamsters don't like that until they are well used to you.
If he's quite tame and shows signs of wanting to come out then get him out when he wants and do any hand feeding out of the cage (in a secure area). He may still need to build some trust with you personally even if he's used to being handled.
First day you could talk gently to him through the bars so he gets used to your voice. I would probably leave it 2 or 3 days unless he shows he wants to come out.
You can spot clean the pee area after a few days (if he has pee'd somewhere other than in the litter tray) and then move the litter tray to that area. Putting a tiny bit of pee'd on substrate on top of the litter tray helps them know it's their toilet area still.
I think they need at least 24 hours to decide where to nest and start hoarding and scent marking and feel secure. If you can keep any of his old bedding from his old cage when you get him, bagged up and sprinkle some of that on top of the new, it may help him settle in a bit