Unfortunately it doesn't invalidate the arrest. In Virginia if the keys are in the ignition and your behind the wheel that equals operation. However, you have some excellent facts for local counsel to start attacking the Commonwealth's case.
If you're convicted you're guaranteed to lose your license for one year. However, the court may issue you a restricted license to use provided you have ignition interlock installed in your vehicle. With regard to jail time it depends on the judge and the facts surrounding your case. The best thing you can do is speak to local experience counsel who can help you find defenses to the case and better prepare you for the outcomes.
You need to file a petition for a restricted license in the Circuit Court of the jurisdiction you were convicted. The forms are available at the Clerk's office or online at the Virginia Supreme Court's Website. There are several steps you need to take so it most likely will be faster and with a better chance of success if you hire local counsel.
This is a jailable offense with serious consequences for your future. If you can afford to hire private counsel, you should hire someone soon. If you cannot afford counsel asked the court to appoint an attorney for you.
A Reckless Driving Charge is a Class 1 Misdemeanor which requires your presence in court. Some local jurisdictions have unwritten rules about out of state drivers that only a LOCAL experienced attorney would know. I would suggest talking to a local attorney in Spotsylvania otherwise plan on being present in Court or face the possibility of having a bench warrant issued for you.