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Protectionist impulses run far too strong on Capitol Hill, especially in an election year.

This is from NewYork Times.
I I barely know what it means, but as an ESL student, I need to analyze the structure of the sentence, in order to improve my English grammar.

I have now idea what the underlined part means and how it works grammatically.

Is it a linking verb? or.. can transitive verb "run" possess an adjective as its object?

Yes, in this case 'run' does indeed function as a copula (or linking verb), with adjective 'strong' as complement.