Cis/trans stereochemistry dictating pKa
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.