There is budget tatami accommodation run by a place called the Kaofeng (Gaofeng in Pinyin I think). Their number is 052679739. Be sure to tell them you want the cheap rooms since the same company has a more expensive hotel just up the road.
There’s the catholic hostel too. 05-267 9602. NT250 for a dorm bed. It’s not always open on weekdays so call ahead.
If you have an extra day consider stopping at Fenqihu about halfway up on the train. It’s much more rugged here and better hiking. Also, the sunrise is said to be almost a minute longer than at Alishan.