Whether it lives or dies, there is no way it will grow back into something that looks nice as a "gift" in ten days, so start looking for another present to give your friend!
Your guess that you overwatered it may be correct. Succulents usually take a long time (months or years) to die from underwatering, or even no watering at all. On the other hand if overwatering causes rotting, that can spread fast.
Turn the pot upside down and try to remove the plant and compost more or less intact. Then scrape away some compost to see what the roots look like.
If they are firm and white, they are probably still healthy and alive. In that case, I would cut off all the "dead" leaves, and see if the plant regrows from the roots. Don't give it any more water until it starts to regrow. Succulents store water in their roots, and that will be enough for it to start growing.
On the other hand if the roots are brown and soft, for practical purposes it is dead, so throw the plant and all the compost away, clean the pot thoroughly to get rid of any pathogens (mould and fungus spores, etc), and start again with another plant.