Type camp sites, stonehaven into Google and you'll find several commercial sites. Bus times are also on line as are B&B, see Visit Scotland. Hope this helps.

If I remember correctly a Challenger being now a stalwart ten timer once enjoyed a sunny wild camp in the immediate vicinity of Dunnottar Castle at Bowdun Head. Hope this helps.

Sorry, I should have specified that I'm looking for "wild" camping options. Not interested in campgrounds. I thought I'd read of challengers camping in the area, but don't know the details. It's really just an option we'd like to keep open if circumstances dictate. Not Plan A.

Kent

There is a small beach area below the castle on the North Side, mostly shingle but you should be able to find a sandy bit big enough for a tent. There is also a bigger beach on the South side (Old Hall Bay). Just follow the track/ cobbled road from the car park towards the castled then follow the clifftop path until you find a suitable tack down the cliffs to where you fancy. The clifftop path goes for miles so you can go further until you find a suitable cove that you can get down into if you want.
The bus stop is on the main coastal road near the junction with the road that leads to the castle. It is all downhill from the castle into Stoney, either following the road or along the clifftop past the war memorial. You can be in the Marine or Ship within minutes on foot, it is a bit of a drag back up the hill though.
During the summer there is a tourist train running from the Market Square to the castle but I'm not sure about when the season will start and what the operating hours will be. https://www.facebook.com/stonehavenlandtrain

Personally I would probably look for a pitch on the South side, most of the tourists just go a little way along the clifftop path to take pictures of the castle, very few venture down to the beach on that side. I wouldn't recommend camping on the cliff top, very little flat area except for the path.

A freshwater burn runs parallel to the road between the carpark and the castle before plummeting over the cliff, you could get fresh water from the burn, or just ask for tap water from the lodgehouse or the catering van in the carpark.

Excellent. Just the info I was looking for. Thanks!

Thinking about it... it is the opposite side of Stonehaven from the castle, but if I was looking for a wild pitch close to Stonehaven, I would head down to Skatie Shore, where the Highland Boundary Fault hits the North Sea.
You head out of town past the golf course and under the railway line until you come to a parking area/turning circle at NO 883877. Across the road a path heads through some woodland and then heads back towards the sea under the railway viaduct at NO 888880. Once you hit the shore, turn right back towards Stoney and there are a succession of attractive bays with a wide grassy area above high water which would make a great area to camp. You would probably want to bring fresh water in with you though, there aren't many streams in the area.