Because Cyprus and Nicosia are divided, you have to cross borders all the time when travelling from North to South. In Nicosia, there are 3 borders: the first one called Lokmacı/Ledra Gate is right in the middle of the walled city and can be crossed by foot/bike only. The second one, Ledra Palace is just West outside of the walled city and can be crossed by foot, bike or diplomatic car. The only one that can be crossed with any vehicle is the Agios Dometios/Kermia/Metehan crossing, and is located much more West from the city center (around 3 to 4km).
Walk on Girne Caddesi outside of the walled city (North Side) through the Kyrenia gate, the one with the statue of Atatürk. Turn left on the big road until the first big cross and turn right on Şehit Ecvet Yusuf Caddesi. After 5min there is a roundabound and take the left (Bedrettin Demirel). There is a bus stop where you can directly hitch from (works well) or continue walking on this road until you feel like hitching.
About 6km later there is a big roundabound that will lead you to Güzelyurt if you take the left. You can walk 5 minutes and have a better spot just a bit further after some traffic lights, before a bridge for people on foot. Or you can go straight facing the mountains and head to Girne/Kyrenia. The road on the right leads to Famagusta/Gazimağusa and is not a too bad spot either.
West (North Part) towards Gazimağusa/Famagusta
Walk on Girne Caddesi outside of the walled city (North Side) through the Kyrenia gate, the one with the statue of Atatürk. Turn right on the big road, which is one-way on the wrong direction. Walk 5 more minutes until the road becomes on the right direction for you. From there you can start to hitch or walk the 3-4km that will bring you to the highway. You might want to walk to the first roundabound that will sort out the first cars!
It is pretty easy to hitch out to this direction. First you have reach the entrance of the highway at the end of Lemesou Avenue. To do so you can try to hitch directly from the city or walk (which should take around 45min). To hitch, try Nikis Avenue or Makarios III Avenue. Be aware that many people will pass by but actually don't go to the highway, and Cypriots being genuine, a lot will insist to make a detour to the highway for you.