this is a loaded question.. Old or not, Garp appears to have one of THE strongest Haki.. can his Haki defend against Mihawk's sword slashes? Garp uses no weapon but his fist.. if you can imagine his Haki-filled fist punching away Mihawk--who is easily 40 years his junior--then Mihawk will have to get creative. If Mihawk is way faster tho, and Garp has no defense for speed, then Mihawk would win cuz he'd be too quick to Garp to defend. BUT, if Garp can defend any body part with Haki Armament, and he can do it quickly.. then Garp wins.. if Mihawk is too quick, or gets creative, or for example he can swing, move, swing, move etc super quick and can hit from 10 different places within 1/100000 of a second each (which is quick lol) then it seems Mihawk may cut him up.. unless Garp can Armament his whole body at once..
problem is.. we don't know the FULL EXTENT of EITHER of their powers/abilities..