What is the priority of a phenyl group in a compound such as this? I assumed this:
http://www.chemspider.com/Chemical-Structure.12491.html?rid=222625d4-811c-4b07-b263-8930fafa831a

to be 1-phenylbut-4-ene
Where is the priority here? No source I know of tells me of where the priority of the phenyl  group lies.

It is wrong. The book states it is 4-PhenylBut-1ene.

What is the priority of the phenyl group?
For preferred IUPAC names, the ring is preferred to the chain.
Therefore, the preferred IUPAC name for the given structure is but-3-en-1-ylbenzene.

However, the context may favour the chain, so that, for example, an unsaturated acyclic structure may be recognized.

When one double bond is present, an unbranched alkene is named by changing the ending to ‘ene’. The double bond is assigned the lower locant possible, which is placed immediately in front of the ending ‘ene’. When several structural features appear, low locants are assigned to ‘ene’ endings before detachable alphabetized prefixes.
Therefore, a correct name for the given structure is 4-phenylbut-1-ene.