Keep going on the highway and follow the same road into the centre.
There is lots of parking signposted around the old venetian walls near the old town. Laiki Yetonia is the old part, and you can walk up Ledra St to see the dividing line. It is possible to walk throough to the North of Nicosia at some point.
It depends what you want to see!

There is a car park across the road from the border crossing. As Carol says I think it is in Ledra Street. We headed for the Paphos Gate and parked up before walking across.

It is possible to take a car across though I've yet to find the crossing point to do this. Locals gave me directions to everyhwere but where I wanted to go.

If you do walk through there are many taxis to take you into Kyrenia for about £10, they'll pick you up as well. You can negotiate a price for them to be your chauffeur for the day and they'll take you to all the tourist spots in the North for about £40.

From what I remember of Nicosia the old wall city is a circle with a number of gates. If you can imagine a clock then Paphos Gate is at about 8 o'clock and if you're entering from Limassol/Larnaca you'd be coming in at about 6 o'clock.

I hope this makes sense. It will when you see a map.

I always head straight into Nicosia from the motorway and keep going until I hit the traffic lights just outside the old town ( you can't go straight on any more and have to turn left at the lights). If you then turn immediately right you'll see a car park which will cost you a quid all day.
It's ideal for getting into the old town and is only a hundred yards or so from the top of Ledra Street.

If you want to cross to the north, keep going past the car park and head towards Paphos gate which is a few hundred yards further on. You'll see a couple of car parks there and the crossing point at the Ledra Palace Hotel is signposted and pretty obvious.

If you have a hire car you'll probably find that your insurance doesn't cover you in the north so you're just as well going over on foot at the Ledra Palace.

While on the subject, a bridge has been built on the northern side of the barricade on Ledra Street (confusing isn't it, with all these Ledra's!) and there's a very faint hope that they will eventually open it up on the Greek side also - but don't hold your breath.

Dave