tl;dr if you want to interpret the rejection of the null hypothesis as evidence that prices for women are greater than those for men, then you do need the assumption of equal variance (in fact, equal distributions) between the two populations. If you are satisfied with showing that the distribution of prices for women differs in some way from that of men, then you don't need the extra assumption.
You don't need to worry about unequal sample size (this will affect the power of the test, but not its validity) or Normality.
For what it's worth, testing whether one group's values are larger on average than another group's when their variances also differ is a surprisingly deep question, even for Normally distributed data (where it's known as the Behrens-Fisher problem).
Referring to the Wikipedia page: the "very general formulation" says:
- Under the null hypothesis H0, the distributions of both populations are equal.
- The alternative hypothesis H1 is that the distributions are not equal.
The next paragraph says:
Under more strict assumptions than the general formulation above, e.g., if the responses are assumed to be continuous and the alternative is restricted to a shift in location, i.e., $F_1(x) = F_2(x + δ)$, we can interpret a significant Mann–Whitney U test as showing a difference in medians ...
Note to technical readers: I think this is a reasonable summary, but if anyone wants to be more rigorous, feel free to comment or edit or post an alternative answer ...