Disclaimer: I've never run one of these reactions, so I don't REALLY know. But, I'll throw this in, anyway.
That reaction is run at 4 atm, which compared to 100 atm is still considered "low" pressure. However, 4 atm is much higher than 1 atm and I think over-reduction would be a problem for the Ph-acetylene -> styrene hydrogenation. Higher pressures are used to increase reactivity (with high enough pressure you can hydrogenate benzene). So, if the reaction already goes at 1 atm, it should go faster at higher pressure. Unfortunately, that would probably be faster than you like and all the way on to ethylbenzene.
I suppose you *could* use that setup at a pressure closer to 1 atm, but the burette setup seems more practical if you're trying to run the reaction at 1 atm. The outside atmosphere does the heavy lifting with pressure control, and with 22.4 L/mol and a burette, precision should be excellent.