I'm pretty sure any old spray bottle that works could do the job, but you'll probably want a small one that releases more delicately. Maybe like people use for perfume? Tiny tiny particles are in that liquid, so no worries about clogging if that's what you're thinking.
I think the advantage when you spray is that you get an even distribution on your felt. And just a few sprays - just a little is needed. Of course it will dry on the felt, but the abrasive particles should remain near the surface unless the felt isn't fine enough. There are a few videos out there on stropping - but I can't remember specifics. You could also use the diamond spray on leather. Maybe take a look? I'm not a strop-expert, though.