We were here in November and all was quiet, so the town had the appearance of an Uthavn rather that guest harbour for facilities. That said we went for a walk around the village and there seemed to be lots to do if you visit in season, including an outdoor amphitheater!
There are two quays you can tie up to in Mosterhamn, the municipal quay to the south edge and the smaller quay to the west side of the bay. The municipal quay had a diesel fuel dock, but not pay at pump so you probably need to arrive in normal working hours to get it (8am-4pm is pretty normal, closed sunday). I assume there is also water here but I did not see.
The other quay had nothing when we were there, but it did seem to be free to use so I cannot complain.
There is a fairly simple approach, especially in daylight. The main harbour entrance is marked and lit with port and starboard lights. To enter the bay boats must pass between the rocks to the north and 2 posts to the south. The first is large and easily visible, the second is slightly small but further away from the middle of the channel. Both have reflective markings if approaching at night.
Once in the bay the municipal quay is to the left (SW) with white painted tyre fenders. The second guest quay is the wooden quay almost directly in front. Depths at both of these are good, around 4 m.
What to do ashore
The town has lots to do ashore, with the outdoor amphitheater open in summer- built into the old limestone quarry. The town has a long history and the church has been claimed as one of the oldest in Norway. There are lots of good hikes near the town, including the mountain Siggjo which is distinctive from sea as Bømlo.
In there a fresh water swimming pond with beautiful clear water and diving boards, this is another use of of the old limestone industry which was once the towns main business.