How much was "Sonnet 130" influenced by modernism?
I think you need to be very very careful with your terms here. Let us remember that this excellent sonnet was written by Shakespeare during the Renaissance, in sixteenth century Britain. Modernism, as a period of literary criticism, actually is thought to have started in the late nineteenth century. Therefore, the quick answer to your question is that this sonnet was not influenced in anyway at all by modernism, because modernism as a philosophical approach had not even come to exist during the time of Shakespeare, and would not for a few more centuries.
However, it is possible to use modernism to analyse the poem. One of the great aspects of literary theory is the way in which we can go back, using the lenses of such approaches as modernism, to look at and analyse texts such as this sonnet and we can see whether this approach casts any new light on the poem. However, we need to be very careful to not confuse our time periods as you have done in this question.