If a person is in the middle of ‘half-Hallel‘ [such as on this Shabbos!], and he is called to the Torah, he should go up, say the berachos , and even read the Torah along with the ba’al koreh (just as he is always supposed to). He should try to finish the paragraph of Hallel on his way up to the Torah, or at least stop at a logical place to stop (at the end of a possuk, or thought). Ideally, he should not be called up (unless he is the only cohen or levi).
There are different rules when whole-Hallel is said.
Does the person sitting next to you in Shul unfortunately chatter and chatter and chatter some more? Consider changing your seat–see Rashi in this week’s Sedra, Bamidbar 16:1, divrei hamaschil ‘vedosson va’avirom’. And so consider it —strongly!