Hi. I should think tomorrow morning would be ok for the vets if you make him comfortable
It does seem like he might need something for his eye, if it's an infection. You could gently bathe it again (sorry just read you already had). Three and a half is a great age for a hamster! He is a very old man! One hamster year is approximately 40 human years so he is about 140 years old!
Just keep him cosy with plenty of bedding and nesting material. He might be low in some nutrients if his teeth have gone bad or broken or dropped out, so you could ask the vet about nutrition as well as his skin and eye. Meanwhile you could try supplementing his hamster food with a bit of scrambled egg and some boiled vegetables (which will be soft). Just a tiny bit of veg each day - maybe a thumbnail size piece.
Not sure what substrate (litter for the bottom of the cage) you have, but make sure there is at least 3 to 4" in there, and put out a big pile of torn up strips of plain white toilet paper in the cage somewhere so he can make a cosy nest (sorry if you already know this). It's the safest nesting material - so if he has any fluffy bedding, then remove it and put the toilet paper strips in. I just take a few sheets of toilet paper and tear each sheet into 5 or 6 strips lengthways.
Assuming his nesting material isn't some kind of synthetic fluff though, it's best not to disturb his nest as that stresses them. Also as he has had a change of environment coming to you, try not to do much cleaning in his cage for a week or two. You could spot clean the odd handful of wee'd on substrate every few days if it's a bit smelly and just replace that with a clean handful. If he hasn't pee'd in his nest then just leave it in tact.
And you could talk to him gently so he feels safe with you