Hi there
I've lived in Limassol for just over 8 years now and I'm originally from UK.
One thing I would say though is not to refer to it as a border as it isn't. You can call it a crossing point or the green line if you want to not upset the local people.
To drive from Limassol directly to Kyrenia will take no more than 90 minutes including stopping at the crossing point, purchasing additional car insurance and completing the slips of paper for the computer system.
The best place to cross for Kyrenia is by the racecourse in Nicosia. You need to exit the highway at the GSP stadium (I think it's junction 4) and probably signed for Strovolos but it's signed for Kyrenia after that until you go through the crossing point at Ayios Dometios, then it becomes Girne.
From there it's well signposted but beware of speeding cameras - they will book you, even if you're a tourist! Unlike here in the Republic, it's unusual for a red plate car to get booked for speeding but there's always a first time!
If you're considering hiring a Z plate (red plate car) to cross to the North from the Republic of Cyprus make sure the hire company you use will allow you to take it through.