Just did this after 2 attempts, the first time I made the awful mistake of using the 'refill' option rather than 'revive' - don't make the same painful mistake!
This can be done by a little luck in the early stages, but I'd advise using a well upgraded robot as that allows you to have a lot more control over the scenario - I used a robot with attack harmony and free attack/defence setups, as that meant I could allow the opponent to hit me now and again but always be ready to smash them away as soon as they had done the right amount of damage. I'd suggest intentionally aiming for 11 health and then allowing them a free full shot at you as the easiest way to end up with exactly 1.
I did it in tournament 4 as there were enough hazards around to try and get yourself down to 1 that way if the situation offered it, but not so many that you're likely to get taken out between getting down to 1 and getting back to the nearest shop.
Just to clarify the above:
Refill - gives you health back 1 by 1 - Don't use this!
Revive - does it all in one go - Has to be done this way for the achievement!