C-4 oxidation of octanoic acid to 4-oxooctanoic acid
If I was tasked with making 4-oxooctanoic acid I would not start from octanoic acid! There is no easy way of making 4-oxooctanoic acid directly from octanoic acid. It is however a relatively trivial exercise from other starting materials. 

My suggestion is to react the commercially available ethyl succinyl chloride (ethyl 4-chloro-4-oxobutyrate) with n-butylmagnesium bromide (also commercially available) modified with manganese(II)chloride/lithium chloride as described in Tetrahedron Lett. 1992, 33 (31), 4439–4442. This will give the ethyl ester of the required 4-oxooctanoic acid. A simple base hydrolysis gives 4-oxooctanoic acid. You could probably use the widely available n-butyllithium in place of the Grignard.