You only need a 4x4 vehicle for the Troodos if you are planning to drive "off road". All main routes into the Troodos mountains from Paphos are on good quality tarmac roads.
Depending on which road you use, the journey from Paphos takes about 1.5 hours so can be easily done in one day. If you do go off road, however, you need to allow extra time as the off road trails are on dusty gravel tracks which wind their way through the vast forest areas. To see the Venetian bridges, for example, you definitely need a 4x4 vehicle.
So yuo can visit the Troodos in one (long) day or perhaps stay overnight to fully appreciate the cool mountain air! Just book your hotel in advance especially if visiting in August as the Cypriots holiday themselves in Troodos at this time and all hotels are very busy.