On May 6, I'm going on a trip to Nujiang Prefecture. I might go from Kunming to Yunlong (in Dali prefecture) first, and then would like to take a bus from Yunlong to Lanping on May 7. Is there direct bus service from Yunlong to Lanping? If not, then what would be the best way to get from Kunming to Lanping?

There is a direct bus between these two cities. There are minibusses traveling along the Pi River as well but these will not go all the way. If you however would like to have a look at the ancient bridges along the way this might be a better option. The main bus probably only stops at the especially for tourist reconstructed rattan bridge convieniently located just next to the road.
Wander why you want to go to Lanping. It is a rather grim mining town without any interesting points.

Good question. Lanping caught my attention because it isn't in the Lonely Planet guidebook. So, I would guess that it's one of the more obscure towns in Nujiang Prefecture (although correct me if I'm wrong). I want to spend a week in Lanping County and go hiking in the mountains. Afterward, I want to go to Fugong for a week and do some more hiking.

Since Lanping is grim, which towns in Nujiang Prefecture (or elsewhere in northwestern Yunnan) would you recommend visiting instead?

You are more likely to find Lanping in a mining guide than in a travel guide. Also Lanping is Nujiang Prefecture it is not a typical Nujiang place. Therefore you will have to go to the Nujiang Valley itself; especially the area around Bingzhongluoxiang is a traditional hiking area.

The Pi River Valley is interesting for hiking but don’t expect to find any accommodation apart from Yunlong, Nuodeng and Jiancaoxiang. Lanping is a mining town and the location of a very pollutant zinc mine.
The Pi River Valley has a wide collection of ancient bridges ranging from metal chain bridges, wooden bridges and even rattan bridges. They are all on the river. For serious hikes you will have to cross from one river to the other crossing the in-between mountain ranges.

OK, I would like to get away from pollution, so visiting the mining town of Lanping might not be a good idea. Maybe instead, I'll go to the Pi River Valley and the Nujiang Valley. I'm especially interested in visiting Bingzhongluo and Dulongjiang.

Please note that May is not a good time to visit Bingzhongluo and Dulongjiang as it is the rainy season. The roads will probably be blocked.

What is the best time of year to visit Bingzhongluo and Dulongjiang? Also, what are some good mountain destinations to visit in Yunnan in May? If the Pi River Valley gets good weather in May, then maybe I can go there instead and skip Nujiang Prefecture.

The summer is the rainy season throughout Yunnan, as we get the tail end of the monsoons, especially where the mountains and valleys are. You can get to your destination most likely, but there could be landslides that close roads and you might be forced on long detours. Best to go now or fall, if your schedule doesn't allow for this, then just bring rain gear and be prepared.

May in general is a good time to travel in Yunnan.
Indeed there is only one single road in the northern Nujiang Valley and that one indeed sometimes gets blocked by mud and / or rock slides. As long as you are not with your own car this is often not such a problem for hikers because most of them can be crossed on foot.
If not you will have an extended holiday ;-)
The Pi river valley can be visited year round.

Note that, over the years, the onset of the (relatively) rainy season, in Kunming at least, varies from about mid-May to early June - no guarantees.