Neuquen can be categorized as the south of Argentina, which means that hitchhiking is really hard.
Hitching out
East toward Bahia Blanca
There is a peaje in Cipoletti 5 kilometres from Neuquen, but you will likely be sent away by the people working there, and don't expect to get a ride.
West toward Bariloche/Zapala
There's an YPF at the edge of town where you can ask drivers, but traffic passes pretty quickly, so thumbing it probably won't get you far. If you can get a ride to Plottier, a town about 10km down the road, there's a stop light right before the highway begins again. If that doesn't work and you can get a short ride, you can try the YPF petrol station in Arroyito.