1. If you have documentation from the Dubai Police to say you have a 1 year ban, then present that to the Dubai immigration authorities to try to lift the permanent ban.
2. Otherwise, the Dubai immigration is the only authority that can lift or change an immigration ban imposed in Dubai as far as I know (or maybe the UAE immigration department - I'm not sure how much autonomy the immigration departments in each emirate have). You could try pleading a special case and ask them to lift it for humanitarian reasons, or someone (family member, lawyer) could act on your behalf. If they go to the immigration office in person, it's probably better than email or telephone contact, and it would be a good idea to try and find someone in authority to speak to - starting at the general information desk is probably going to be unsuccessful. If the immigration ban is due to a criminal conviction then there's little point in trying to explain to the immigration department that you are innocent. I doubt that they are in a position or have the desire to judge court cases.
3. Alternatively contact a UAE lawyer to see if there's any way to appeal and/or overturn the conviction. Sorry, I'm not familiar enough with UAE law to be able to say how realistic that is, or if it is even possible.
4. Regarding the confession, there are Human Rights Department offices at Dubai Police stations
although I don't know how far you'd get given the amount of time that has elapsed. It might be more useful to contact a lawyer to act on your behalf if you want to address the matter, or to at least give you some information on how realistic it is to pursue it. Some telephone numbers you could try (from the Dubai Police website):
Tel +971-4-6082629 - General Director of Department of Human Rights Care
Toll-free (in UAE) 800-404040 - users complaint section
Tel +971-4-6082305 or +971-4-6082304 - public complaints section
The Dubai Police operator at General HQ is tel +971-4-6099999.