That runout in and of it's self won't bother you a bit. You're still inside a center punch, the drill will find it's hole.
Drill presses, being "not milling machines" don't have the rigidity to force a cutter to stay on center. Nor do they have the rigidity to force a cutter off center. Nor are most drill bits in that drill's vocabulary rigid enough to care how rigidly they're held. They'll bugger off on their own just as easy on a mill as they will on a drill. You also might find that if you started measuring different points along your drill bits, they might not be as straight as you'd like to think they are anyhow. Don't go getting the micrometer back out.... You were already good with that tolerance, the world is OK with that tolerance. They worked fine up until now.
Basic rules for drill pressing things:
Center punch critical work.
If you you have a particular material(s) that you have a hard time getting a drill to start "riight where it lands", center punch that too.
If you MUST use a jig or fixture to locate holes in an "unmarked" condition, start the hole SLOWLY. Hard on drills, but just like drilling on a lathe, if you start slowly and gently, the drill bit will find it's center. Initially anyhow. They'll almost always wander on deep holes, but there's pretty good workarounds.
If that little bugger is big enough to do the work you've got lined up for it, then set it up as best you can, bolt it down to something heavier than it is, and use it. Drill presses inherently have metrology issues, it's just how they are. Use the drill press. Address performance issues. Preferably without micrometers or dial indicators. If the runout is any more than just barely visible at the tip of the drill bits you're using, before you get out the dial indicators, try clocking the chuck in a couple of other positions about 120 degrees from where you started. Yeah, I'm guilty too, but measuring drill presses to thousandths of an inch is best saved for chasing down where damage occurred, and how to fix it. They're pretty floppy overall, even the big ones. So having it more accurate than it's flexyness and overall cast in and, machined in "tolerance" is pretty much not going to get you any further ahead. STOP! Put the micrometer down. You don't wanna know about that stuff either. Use the drill press and see what it does.