What kind of procedures would be necessary to undo the legal death?
What happens to his property (note: the reanimated character is a
young man with no written will)? If he's a renter, how long will it
take for his landlord to push all his stuff out and welcome a new
tenant? How much paperwork does he have to fill out?
If the person is on the autopsy table, while he may have technically been legally dead, the person's death certificate has probably not even been issued yet, since normally an autopsy is done, in part, to determine a cause of death for purposes of preparing a death certificate.
Normally, an autopsy is only prepared when the cause of death is uncertain or needs to be documented for legal reasons (e.g. an accidental or potentially criminal death or one that is just unexplained), and the need for one postpones the preparation of a death certificate. If the cause of death is self-evident and there is no indication of foul play, an autopsy isn't done and the preparation of a death certificate proceeds more quickly.
In other words, while he may be "legally" dead (defined as discontinuation of the function of certain bodily processes like brain activity and heartbeat and breathing), he very likely isn't yet "bureaucratically" dead.
You can be "legally" dead days before anyone even discovers that you have died, with the time of your death determined only retroactively, but that doesn't have any practical effect until someone discovers it and it makes its way through the bureaucratic and court process.
It takes a few days after a death for a probate proceeding to establish death, heirship and the validity of any will (many states have a 120 hour minimum waiting period after a date of death and most autopsies are within 72 hours). Even if a probate proceeding had been commenced, a simple affidavit or notice that someone was not, in fact, dead would cause the case to be summarily dismissed and it is unlikely that much would have happened in the court process since then.
With older people, one of the fastest agencies to act is the Social Security Administration discontinuing benefits. But a younger person wouldn't have gotten those to start with, so that wouldn't be a factor.
Life insurance companies normally won't pay until a final post-autopsy death certificate with cause of death is in place (so they can look for fraud in the life insurance application relative to the cause of death stated).
It would be very usual for a landlord to take action sooner than the month after learning of the death, and often the estate will keep in a lease in force much longer. Next of kin may have removed some valuables prior to any official declaration, and while most people would promptly return them upon learning of the "mistake", a few items might not be recovered.
The final death certificate would probably never be prepared. A preliminary death certificate might be quashed as erroneous and not filed with the Vital Statistics Department of the State Department of Health, by the preparing party, through a simple, oral, in office instruction to the person in the coroner's office who handles those filings.
Since there isn't a formal judicial declaration of death or a final death certificate, simply informing an agency or person that received a preliminary notice that it was in error, perhaps by letter or email, would suffice. Even if a final death certificate had just issued, the person who filed it could probably fill out a simple correction form so soon (probably just a matter of hours) after it was filed. It wouldn't be that disruptive.
The messy cases are when there has been a legal declaration of death and then months of actions have been taken in reliance on the determination that someone is dead.