Merc,
In my limited experience, 1903s are fairly straightforward rifles. At the risk of stating the obvious, make sure everything works. Check the operation of the bolt, the safety, the magazine cutoff, and the rear sight (the sight ladder should snap smartly into place; the windage adjustment should be smooth but not loose). If possible, take along a snap cap and dry fire it a few times to feel the trigger pull. Also take along five dummy rounds to make sure it cycles properly through the entire magazine. (I've seen feeding problems with these in the past that were the result of improper receiver geometry.) Look for stock cracks and stock fit - does the trigger guard fit snugly against the stock? Look for any signs of cracking behind the receiver tang. Also, carefully check the bolt lugs. These are pretty much the same things I'd look at on any older bolt action rifle.
Keep in mind that, if the barrel and receiver are good and headspace checks out, just about any other problem can be fixed - sometimes cheaply (weak magazine spring), sometimes more costly (cracked stock). Small problems can sometimes provide you with considerable bargaining leverage. HTH. Good luck!
"They've took the fun out of running the race. You never see a campfire anywhere. There's never any time for visiting." - Joe Redington Sr., 1997