Do tubed tires fit all rims, or is there anything I should watch out for? My bicycle is about 25 years old and for the past decade has been sitting out on my balcony (dry, but outside temperature, which around here means from -15°C to +40°C or so). Lately I've been thinking about reviving it. Once I take it out for a spin I'll know what else I need to fix, but for now the obvious thing is the tires, which don't hold air at all anymore.

I wrote down the size numbers I found on them - 26x2.125 - but I don't know if they are tubed or tubeless. I suspect they are tubed, but I won't be able to tell until I dig the thing out. And it would be nice if I had the replacements ready by then.

So, my question - If I go into my LBS and buy the first tubed tires that I see with matching numbers on them - what's the risk they won't fit?
If it's 25 years old, it's a tubed tire. 26x2.125 is a very standard MTB tire from that era. If you get any MTB 26" tire in the 2.2 to 1.5 in range that should work just fine on your bike. 

For any tire made in the last 20 years there should be an ETRTO number on it as well, this is a number like 50-559. Its' designed tire width and rim diameter. So 26 x 2.125 would be 26 inch ( or 559mm ) rim and 2.125 inch width ( or ~55mm width).

I would put some money in your budget for new tubes as well. 

For more on tire sizes check out Sheldon Browns Tire Sizing Guide