I can help you out here. I've rented a BRZ from Ren4Ring previously and have put on 50+ laps around the Ring now over the past 5 or 6 years.
I do not believe they still rent them out - as I rented this one a few years ago it was their last one due to drivers totaling the fleet. They are a great outfit though and if you're ok with going for a smaller hatch their Swifts are fantastic and reasonably priced. You'd be at about 389- euro for 4 laps all inclusive. Go on a weekday if you can, the Ring will be open for about 2 hours for touristenfahrten; check the calendar for days/times:
https://www.greenhelldriving.nuerburgring.de/#/landing click on the German flag in the upper left corner for English, then Opening Hours Nordschleife.
Weekends in the summer are rough, lots of drivers and lots of closures due to accidents.
Other outfits I know of to rent cars from:
http://www.rsrnurburg.com/car-rentals
Very pricey but the most known and biggest fleet of cars.
https://www.rentracecar.de/en/
They have a GT86 to rent looks to be about 649- Euro for 4 laps all inclusive.
As a first timer, 4 laps will be tiring but you can generally fit in 5-6 laps in a session provided there are no accidents. I wouldn't advise that - check out the parking lot, grab some merchandise, take it all in and do a sighting lap, do a leisure pace lap, then a fast lap, then a 'remember this' lap. If you still feel up for it, take another spin but don't push yourself and drink plenty of water.
Personally, I take the lower insurance deductible (costs extra) because it's a numbers game and with first timer Xbox confidence those numbers go down for you. It's also around $100 per meter of armco all said and done if you have contact with any part of the track. Then you have to pay for emergency vehicles, truck rental for repair, safety car, etc. if needed. Plus state tax on everything
So don't have an off if you can't help it, it'll easily cost you a few grand out of pocket.
My best tip: watch as many youtube crash videos as you can before going. See where the mistakes happen most often. There's a few notorious trouble spots: keep a cool head at Schweden-kreuz, watch out for Kallenhard - it's a sharp turn same goes for near Breidscheid, and go easy through Brunnchen; lots of folks show off for the paparazzi there. Most cars with semi-slick rubber and sub 200 hp will handle the course with little necessity for braking. It's driver courage and competence. You can go virtually 90% flat out in the Swift if you're on the right line and still run a sub 10 minute lap. But don't time yourself - you're racing for points that no one cares about and that mean nothing to you unless you go back regularly. If you'd feel more comfortable take a passenger with you to help you keep an eye in your mirrors. No point by's, use your indicator and keep right to let the faster vehicle pass on the left (German street laws apply to the Ring). Don't slam on the brakes to let people pass. And always remember to look for motorcycles!
My family is from the Trier area, about 40 min out. If you're into Roman history or Marxism it's a great city to visit.
Forgot to say, if you have any other questions about what to do/see in the area feel free to shoot me a PM or post here - I'm always happy to share what I know.
When going to to the Ring, go early in the day to do a tour of the museum if open - lots of cool race cars. Eat something, walk the Boulevard (boring) and be sure to show up early to your car rental spot before the driver brief to get the nervous dump out of the way, otherwise you'll be waiting in line! :p