"To science the sh*t out of something"

The rhetorical term for the phenomenon you describe is catachresis.

Catachresis has been defined by Robert A. Harris as "an extravagant, implied metaphor using words in an alien or unusual way." A noun, for example, could be used as a verb, as in the case of "I'm gonna have to science . . .." Harris gives an example which is similar to Matt Damon's sentence:

The little old lady turtled along at ten miles per hour.

A good catachresis creates a word picture. The sentence which contained the word turtled as a verb may solidify, so to speak, the writer's or speaker's idea in the mind of the reader or listener.

The image of an old lady turtling along cements itself in someone's mind more readily than a mere verbal description such as, "The old lady plodded along slowly" (although the word plodded is better than an average word, such as walked).

In conclusion, while the expression "to science something" may not serve to create a word picture, that one word may get a listener to think of the elements of the scientific method (e.g., the control of variables, careful measurements, and hypothesis-making and hypothesis-testing). Now that's an efficient use of words!

This question can be approached as one of register.

First, there's an expletive used as an intensifier:

beat the crap out of...knock the hell out of...kick the shit out of...punch the bejeezus out of... etc

He knocked the hell out of Joe Schmoe in the heavyweight bout last year.

Second, there's the use of the noun science as a verb, occupying a particular position in a common intensifier pattern, and thereby associating doing scientific stuff with punching, beating, kicking, knocking, etc. Using the noun as a verb is like using a wrench as a hammer. These two things, noun as (crude) verb, and noun-verb occupying the position of kick, beat, punch in the intensifier pattern, turn his scientific activity into a macho or survivalist enterprise. He is momentarily feigning the register of the macho survivalist.

There's self-deprecating humor (lab geeks laughing at themselves) but also a real swaggering pride taken in his possessing the knowledge that may save his life, the kind of knowledge that could save his home planet's life some day.

So, when the OP asks for a "better" way to say this, the question arises, what does "better" mean? A more polite, or a less swaggering way? That wouldn't be better from the point of view of the story and character. A neutral way would be to say that he intends to use every scientific method at his disposal.