Tubulars or clinchers won't affect your ability to spin, at least not in the range Zapp is looking for. Fixed gear on the road is useful, but not required. In fact, sometimes people are afraid too spin too fast with a fixed gear on the road because they know they can't just coast if things get a bit hairy.

Better to use the normal road equipment and spin up to 120+ and try to hold that against low resistance (down a slight hill) for 10+ seconds. Then work on increasing the rpm's you can hold for 10-20 seconds. Sprints on the flat for 10-12" out of the saddle in a 53x17,18 are good practice too.