Is there a train connecting Lijiang to Shangrila? or the bus is the only mean to reach it?

Oh, no train connects these two places.
You need to take direct long-distance buses to arrive there. The buses at Lijiang Bus Station depart from 07:30 to 17:00 with the interval of around 40 mins. And the ticket fare is CNY 63 per person. Generally speaking, it takes 4 to 5 hours on the way.