Vinyl benzene is a common starting material and intermediate in second semester organic chemistry synthesis problems, so you should be familiar how to make vinyl benzene from benzene (see problem 722).
The trick to many synthesis problem is to realize how many carbons need to be added, as that will determine which route you use.
a) Adding zero carbons: alcohol then oxidation
All you need is an alcohol added to the last carbon, which you can oxidize to a carboxylic acid. So from the alkene do an anti-Markovnikov addition of H2O via hydroboration, followed by oxidation using Jones reagent (aqueous chromium, Cr6+ in acid).
b) Adding one carbon: nitrile (cyanide, CN)
When you add one carbon that screams "cyanide!" The most common way to add CN is using NaCN with an alkyl halide in an SN2 reaction.
Because you want to add the extra carbon (CN) to the end of the chain, you want to do an anti-Markovnikov addition using HBr and perioxides. Then add NaCN to form the nitrile.
To convert a nitrile to a carboxylic acid, hydrolyze under acidic conditions (add water with acid, H3O+).
c) Adding two carbons: epoxide
When you need to add two carbons, ethylene oxide is the way to go. But ethylene oxide is an electrophile, so first you must transform vinyl benzene into a nucleophile, specifically, a Grignard reagent.
We want the Grignard part (MgBr) on the end, so do an anti-Markovnikov addition with HBr/peroxides followed by Mg0/ether. Then add the epoxide which adds two carbons with an alcohol on the end. Finally, oxidize using Jones reagent.
Each carboxylic acid can be converted to its methyl ester via Fischer esterification: add methanol under acidic conditions (CH3OH/H+).
