Learn English – What does “couldn’t happen to a better person” mean

Well, I bit the bullet and asked the guy what the heck he meant (imagine that). Turns out he was referring to the oft-quoted teach a man to fish proverb.

He also clarified that this is not a common expression, just an idiosyncratic reference he invented and uttered spontaneously.

Henry's answer is exactly right. Here is confirmation:

A clearer, more complete use of "cutting" comes later in the same paragraph.

He wasn't willing to cut a subsidy for heating bills.

This illustrates that "cutting," as used in your example, had an object (parts of the proposed budget), and was not just used as an expression meaning something else.

The sentence in your question might be paraphrased in this way, for the sake of this explanation: "As the government programs fell and the tens of billions in savings piled up during the budget debate in the House last week, freshman Republican Frank Guinta joined fellow congressmen in reducing spending and removing parts of the proposed budget.