Ack!
I say: yes, you can tune up your Supra, and days keep track of your time spent playing GT1. A day passes when you participate in a race or complete a License test. (resetting in between excluded)
Tuning is a way of life here in GT, and you can even enter in a series devoted to Hard-Tuned cars. Be careful, though; if you add racing modification (a new paint scheme) to your car, it becomes ineligible; the series is reserved for stock-appearance cars.
Before you upgrade it though, you may want to try the Normal Car series, reserved for cars literally 'straight from the box'. Do not touch the car, and you will be eligible here.
In short, both events are a lot of fun, and they pay A LOT OF CURRENCY in the end if you succeed.
Hope you can benefit from the information!
And, welcome to the fun-filled world of GT Planet, vp3000.
EDIT: Sorry, daan. I guess I should have checked this, huh?