How should I nail blocking?
Since you have a palm nailer, use that. However, I think it is worth mentioning that in lieu of a palm nailer, an electric screw driver, (which is more commonly owned by DIY folks than a nail gun), is easier to use in this situation than a regular hammer and nails.
Oh, and the way to nail them in is to stagger the blocks.

Edit: Please excuse me; for some reason, I thought you were having space issues (I misread). As everyone else said, use the regular nail gun.