I've got the Alaska myself for my Syrian Milo and I personally find it a bit on the small size for him and am going to upgrade him to a larger DIY cage (either 200x60 or 150x50cm). If you don't want a cage that big (which I can imagine :P) the Barney and Alexander cages are great choices, both at approximately 100x50 which make a good cage for any species of hamster.
Again, the Alaska probably will be fine for a Syrian, but I recommend going for an Alexander or a Barney as the extra space you get from it likely is so so nice for decorating the cage and adding in more toys.
Also, I can confirm that upside-down shoeboxes (or in my case ice cream boxes. :P) work wonders as houses. I gave one to Milo and he dug his way down to create tunnels and rooms and it was great, man. Highly recommend that as an alternative for a house you have to buy.
I'd toss out any wooden stuff you might have if you don't know the cause of death of your previous ham (I'm sorry for your loss. :c), but if it wasn't contagious, it could be fine to sanitize it in boiling water but I'm not too sure on that, to be honest. Drinking bottle and food dish won't need replacing. :P
Other than that I think everything was covered.
Good luck and have fun!