Aside from any physical limitations he may , which should be addressed by his Vet- maybe he is not motivated to go up? Have you tried offering him treats to go up? Throwing his favorite toy? Shih tzu's have short legs but are very capable of climbing stairs-
I recently watched a Dog Whisperer episode where a bull terrier didn't want to climb stairs and go for a walk- he was not motivated to do so- as soon as Cesar gave him the motivation , which was another dog to follow- he ran up those stairs and went about his walk happily.

Actually, it's scarier for them to come down steps.

Give him more time to overcome any fear he may have. When he is motivated enough, you won't know he ever had a problem. It took my rescue pug about a year to come down the stairs but now, it's a snap.

Are the stairs carpeted or hard surface? If hard surface, has he ever slipped on them? Small dogs can have problems on slippery hard-surface stairs.

Two things come immediately to mind, that his eye sight might be compromised in some way or that his joints may be inflammed, maybe he has hip dyplaysia. I would take your pup to the vet to rule out ANY physical issues. That said dogs can have quirks, my dog wont eat his food unless you pretend to steal it. Someone must stand at the bowl and grab for his food and threaten to take it from him, or he wil not eat (now how weird is that)!!! Please take your pup to the vet to rule out any physcal problems first. I would love for you to repost afterwards to see what the vet says, and if it is not a physcial problem, then we can help you tackle the behavior.