Naming of a Hydrocarbon with a Benzene Functional Group
First, by your logic, it would be 3-phenylbut-1-ene. Note that the 1 is right before the ene, indicating that is where the double bond is. It is a perfectly understandable name, one that many chemists would use for this compound.

The other way to name it is to consider it a substituted benzene. Then you get (but-3-en-2-yl)benzene. This is the preferred IUPAC name for the compound, because benzene is the base molecule by IUPAC priority rules (rule P-52.2.8 of the 2013 edition of the Blue Book: “Within the same heteroatom class and for the same number of characteristic groups cited as the principal characteristic group, a ring is always selected as the parent hydride to construct a preferred IUPAC name.”).