"Don't try and Lance Armstrong your way outta this one!"

I'd interpret this as a newfangled way of saying:

"Don't try to weasel your way out of this one."

As a verb, to weasel means to obtain some sort of personal gain through questionable behavior. Some dictionaries also list weasel out as a phrasal verb; Collins defines it as:

to evade a responsibility, esp. in a despicable manner

As a noun, a weasel can be a devious or sneaky person.

So, in the wake of the doping scandal, Lance Armstrong is an outed weasel. He denied allegations for several years, but that was found to be all lies. So, substitute "Lance Armstrong" for "weasel," and we've got the quote you presented to us.

I don't think it's an establish idiom yet, but it has some potential.

A common phrase is "Don't lie your way out of this one" or "You can't just think your way out of this one", which mean "Don't lie in order to resolve this situation" and "Just thinking won't resolve this situation" respectively.

The writer verbifies (i.e. using a non-verb as a verb) "Lance Armstrong" to mean "act in such a way that lance armstrong would act." This is pretty ambiguous, but this could be referring to him being accused of taking performance enhancing drugs and him denying it.