Had he lived, would U.S. President James A. Garfield have won re-election in 1884?

It's impossible to answer this question with any great degree of accuracy for the simple reason that Garfield only served as President for just over six months before his assassination. So one can really only speculate. On the whole, one would have to say that it's more than likely that Garfield would've been re-elected. He was generally quite a popular President, both with the members of his party and with the American people as a whole.
On the other hand, Garfield's margin of victory had been quite narrow, just a 0.1 per cent margin in the popular vote. (Though Garfield's victory in the Electoral College was more convincing it was still less than impressive.) The closeness of the election was generally down to a backlash against the rampant corruption of previous Republican Administrations. One such scandal, the Crédit Mobilier bribery scandal, had involved Garfield himself, and his Democrat opponents made great play of this, contrasting what they saw as Garfield's dishonesty with the unimpeachable integrity of their candidate, General Winfield Scott Hancock. As it's more than likely that similar scandals would've plagued a full Garfield Administration, then it's by no means certain that he would've been reelected in 1884.