In talking to the Tibet Mountain Guide School, this mountain goes by two other names. These are, roughly translated into English phoetics: Jambo oozer (Gampe Utse) and Drepung oozer (Drepung Utse). Nobody I talked to seemed familiar with the mountain name Gyalpheri.
1) don't ask the monks nor the locals for access information. Chinese people will give you any old answer when they don't know how to respond to your question. When asking locals and monks which peak was Gyalpheri, I had responses which covered all visible peaks in the area. One even told me to go downhill to climb the mountain :).
2) You don't have to pass through the monatery nor pay 55 yuan to access the trail.
Take bus 301 or 302 from Beijing Dong/Zhong Lu to Drepung. This costs 2 yuan. The bus will drop you off in the village at the base of the road to Drepung. You can hop onto another bus/car or hike the 20 minutes up the road. If you hike, in about 10 minutes you will come to a fork where you would go right to Nechung Monastery. From here you can go toward the Nechung Monastery a bit and then take a left onto the well-traveled dirt trails up toward Drepung Monastery. This way you bypass the ticket office and see more prayer flags.
There are trails and prayer flags all over this mountain, which can cause a great confusion about where to start this hike once you have found Drepung Monastery. I think the Gyalpheri trail is best started from the Drepung road, about 100m before it deadends in the monastery itself. At this point there will be a well-worn dirt path heading uphill toward some painted rocks. Follow this path up the hill and pass all the painted rocks. About 70 feet past a small temple, turn right onto a prayer-flag covered trail.