I must admit I have not been down to Mikros Asselinos, as its a bit of a hike down from the track, but I have driven past the track down to it many times. I believe there is a taverna, but don't know what other facilities there are (sunbeds etc.).
To get there, it is easiest to go from Troulos and head towards Meghalos Asselinos at the Y-junction in Troulos. before you get to Meghalos Asselinos, take a right fork heading towards the dog santuary and Ekklisia Panagia Monastary. The road here turns into a dirt road, but it is quite a good track for a vehicle. After 1-2 miles, you will see a sign for the beach. I think you have to park up at the top and walk down. As access is via a dirt track, it is best to do it in a 4x4.
If you carry along the road you can get to Ligharies and Kechria beaches, but the track gets narrow and rutted the nearer you get to them.