There are several ways you can travel to Epulu from Beni. Flying, renting a car, hiring a motorbike, taking a bus, or riding on top of a truck are all possible options.
The fastest, and most expensive, is by flying. MAF Congo offers flights to Epulu at about $500 each way, which is for the entire plane. I believe the flight time is only an hour or so. There is not a set schedule, and there is a possibility their planes will be in use elsewhere the day you want to fly. Contact MAF Congo for the latest prices and availability.
For about $100 per day, plus fuel, you can rent a car with driver in Beni. The best way to find a car is to simply ask around town. Prices will vary, and a little negotiating will be needed to get a good price. There is an internet cafe next door to Hotel Source Kabungulu (outside the hotel grounds, not the one directly inside the front gate), and the owner there offered us a car for $300 + fuel for 4 days. With the road in good condition and no delays, driving straight should take approximately 6 hours via the Komanda-Mambasa road. The road from Beni to Mambasa (bypassing Komanda) is in bad condition and not a good idea to drive on.
Motorbikes can also be hired. Drivers will ask for about $100 each way, per bike, though this can be negotiated down to about $75 each way. A price of $300 round trip should include fuel and their expenses while waiting in Epulu (granted you are only staying for a couple days). Drivers who work the local Beni motorbike association will need special paperwork to enter the Orientale province. Make sure your driver has proper paperwork, otherwise you will run into problems at the provincial border. Driving time is approximately 8-9 hours, without many delays (again, on the Komanda-Mambasa road).
There is bus service from Butembo to Nia Nia, passing through Beni, Komanda, Mambasa, and Epulu. It is run by Les Enfants D'Abord. I believe the bus normally starts in Goma, but at the time of visit and writing the road was closed due to the ongoing conflict with rebels (hence the bus starting in Butembo). Unlike the other options, the bus takes 2 days to reach Epulu...spending a night in Mambasa if it crosses the Ituri River before 5pm, or Komanda if it does not cross the river before 5pm. Travel time from Beni to Komanda is approximately 4 hours, and Komanda to Epulu 9 hours (including the time to cross the river). The cost is $25 per person, each way (which is actually the same price to Nia Nia). Budget a few more dollars for the night spent in Komanda or Mambasa. The bus station in Beni is located on the side of the gas station on the southeast side of the main traffic circle where the main road meets the road to Kasindi.
Finally, it is possible to sit on top of a truck loaded with goods. I met one local man who claims he traveled from Nia Nia to Beni in one day using this method. Traveling this way can be dangerous, though. If the truck tips over, and you're on top, it will be a hard landing. Prices should get generally cheap though...less than the bus. Travel time, though, is very slow because of the trucks being overloaded, stopping often, and breaking down.
For any of the road options, keep in mind that the Ituri River crossing in Komanda closes at 5pm each day. A car or motorbike should only take 30 minutes or so to cross. A bus or truck, loaded with goods, takes over 4 hours. Also, if it rains, the roads can become quite a nightmare, and travel time will definitely be longer.