Why is trans-4-phenylcyclohexane-1-carboxylic acid a stronger acid than the cis isomer?
In the cis-isomer the carboxyl group is axial; and, given the 1,4 distance from the phenyl, why would solvation be any more difficult?
The bulky phenyl group will preferentially occupy an equatorial position. Therefore, the trans acid has the carboxyl group equatorial and the cis acid has it axial, as you have rightly pointed out.
The difference in acidity is to do with the resulting solvation of the carboxylate group. The negatively charged carboxylate group is more strongly solvated by water molecules, and has a larger effective steric bulk, than the neutral, protonated form.
So, there is an increase in the steric bulk of the group upon deprotonation. Such a process would be more favoured if the group is equatorial, i.e. the trans isomer is more acidic.
Of course, in the absence of pKa values, such an explanation is only theoretical in nature. Solvation is a very difficult concept to quantify objectively.
A possible reason has to do with the need for the ionized carboxylate group to be solvated by water molecules--lowering energy by delocalizing its negative charge. More water molecules can contact/hydrogen bond with/solvate the carboxylate group of the trans isomer than is possible for the cis isomer, due to the phenyl group being placed closer to the carboxylate in the cis isomer.