I'd just pick up a new Zerk or two next time you hit the parts store.
They make a cleaning tool for them but my thoughts are for a few bucks, you'd be all set. Besides I have too many specialty tools as it is.
NorthernTool.com offers a set of 100 for $20.00 (US) but be careful, the only ones I saw there were SAE. A couple from the stealer shouldn't set you back too far.
As far as driving, you'll be fine for a few days.