It's not mentioned explicitly in the book, but most probably neither there was a "back door" nor Dumbledore had to actually "conquer" the traps.
Firstly, the traps were created by Dumbledore's employees on his request. So, he could simply ask them to neutralize their obstacles while he was installing the Mirror.
Secondly, the book doesn't mention when exactly each of the obstacles was installed. We know for sure that:
- the Stone itself was moved to Hogwarts on July 31st,
- Fluffy was in place in September,
- the Mirror of Erised was installed as security precaution around Christmas.
But we don't know about the others, so it's possible that they were added after the Mirror.
Lastly, the obstacles where not realy hard to overcome, as three first-year students could defeat them. Dumbledore (believed to be the most powerful Wizard in the country) would get through them easily, especially if he knew what to expect.