How can I synthesize phenylpiracetam from styrene or benzaldehyde?
This is a reasonably sound route. I have some observations:
The prep of nitrostyrene from benzaldehyde is well known, it is best to use freshly distilled benzaldehyde. Here are the conditions from org syn prep.
The sidechain can be added by treatment of the phenylpyrrolidine with $\ce{NaOMe/MeOH}$/Methyl chloroacetate then heating with $\ce{NH3/MeOH}$
(patent).
You are better off reducing the nitro group at the diester stage as it will spontaneously cyclise to the pyrrolidine. On hydrolysis it will then decarboxylate. In my experience, hydrogenation of alkyl nitro compounds does not give good yields. $\ce{Al/Hg}$ amalgam is better. Sodium hypophosphite /phosphinic acid with cat. 5% Pd/C in water/THF and ultrasound was recently reported here as an efficient method for reduction of alkyl nitro, and the paper contains an example of spontaneous cyclisation of the product amine onto a methyl ester.
A synthesis of phenylpiracetam is described here in this patent.