Depending on your situation, you may or may not get to check out your bus in person before you buy. Obviously, getting to look at it, and even test drive it is ideal...just not always possible. If you can check it out, here are some basic things you can check.
If the tires are extremely worn, or have visible cracks on them, that's a dead give-away that you'll need to replace them. But more often than not, the bus you'll be buying will have tires that look okay, but might STILL need replaced. The general rule of thumb is not to run a tire past 10yrs. So check the date code on the tires (this is especially important on the front (steer) tires, since a blow out on the front is tricky to control. If you just need to get the bus home, and the rear tires are over date, but still in good condition, most people would just cross their fingers and go for it. Replacing tires ranges from $250-500 apiece, so budget for this if the tires need replaced.
To check the date of the tires, find the 4 digit code (might be on the inside of the tires, so you'll have to crawl under the bus with a flashlight). If it's a 3 digit code, then the tire was manufactured before 2000, so replace it. The first 2 digits of the code are the week it was made, the last two digits are the year.
Depending on where you're shopping for a bus, you may or may not be able to be picky about rust. In California or Oregon, you can find completely rust free buses. In the midwest or north-east, that's extremely unlikely.
In any case, crawl under the bus, and look for any significant rust problems. Surface rust is okay, but if it looks like major rust on the frame, you'll want to pass on the bus. If the frame is fine, but there's some major problem spots on the body, and it looks like more than just surface spots, you still might want to pass. Once you start pulling up the floor, you'll likely find even more problem areas. These can usually all be patched up, it just depends on how much time you're willing to spend chasing rust problems.
3. Water in air tanks
If you've got air brakes, start up the bus and let the air tanks fill up. Then shut off the bus, and pull the lever/cord on/under the tank to drain it. If there's a small spray of water that's fine. But if there's a LOT of water, that could indicate a more serious problem, and that the bus hasn't been well maintained.
Check to make sure all the lights, turn signals, etc work. Chasing down weird electrical problems can be a headache. And if the batteries are dead, budget for an extra $50-200 for new batteries.
5. Engine/Oil issues
Start up the engine, does it smoke continuously? Just a bit at startup (usually not a big deal). Black smoke? Might just need a new air filter White or blue smoke? Could be a bigger problem—injectors, blown head gasket, etc. Check for signs of major oil leaks (a little leak from an old diesel engine is probably pretty normal. Check for signs of significant oil leaking from the tranny and axles as well. Check the coolant as well—any oil in the coolant is a bad sign.
And remember, that even with all the checking, you're still buying an old bus. The day I picked up my bus I paid $1,000 for a mechanic who specialized in my type of engine (Detroit 2 cycle diesel) to do a full inspection/tune-up/full fluids replacement on my bus. Eight hundred miles later, I had to replace the engine ECU to the tune of $2,000. Obviously, that's almost a worse case scenario, but be prepared for the unexpected!