frankly this is VERY hard to do in a believable manner without modifying the way reproduction works. This comes down to evolved behaviors which have been built into a psychology for thousands upon thousands of years. A simple culture change is going to have a difficult time overriding our ingrained evolutionary instincts. Those instincts are for men to woo and/or objectify women with a desire to have sex with as many as possible, and for women to be selective about who they have sex with. These are partially part of our evolutionary psychology.
Men stopping objectifying women, that is believable and desirable (and I hope will mostly come to pass over the next generation or two). However, women choosing to objectify men is harder. For that to happen women must feel they need to fight over men as mates, effectively that males are the harder sex to get to agree to reproduce. Considering how little it 'costs' a man to have sex (at least some men, I disagree with those that would father a child without thought of the welbeing of said child, but sadly those type of men are not in short supply) it is hard to believe that women would ever be fighting over getting a male as a mate.
Without the need to 'fight' for men some of the above, like buying drinks just to have the opportunity to start talking in hopes that maybe something will come of it, seem excessive. You basically need a system where getting male mates is as hard in your culture as getting a women to choose to be your mate is in our current culture, despite the fact that women pay a MUCH higher biological burden in raising children and thus have a much higher incentive to be selective AND a genetic predisposition to be selective which will still be present even if cultures change such that there was less of a burden on women (evolutionary psychology gets built over tens of thousands of years, it doesn't really 'catch up' with cultural changes that have been around for a mere generation). In other words it must be harder to get a male mate in your culture, and cost far more, then it is to get a female now; since your competing with biological predispositions which lean towards selective women and precocious men.
...at this point I feel I should stress that I'm talking about averages of huge populations, I stress that individuals can vary widely from this average, but culture is based on the 'average' and thus that is what were considering. I want to stress that it's usually dangerous to take any of the generalizations I use about averages and try to apply them to any single individual, male or female, and presume them to be true. Many will not fit these generalized statements.
As such the best way to believable make this happen is to work with the reasons we evolved the way we did before. Right now women pay a greater cost and are less available then men for reproduction. You need to culturally change these factors.
Phillip gave some good ideas for how to do this. Making men ultimately pay a significant financial cost for children would go a long way towards making them the sex that is selective and needs wooed; though I point out this only works if you know paternity as a certainty AND can find the man easily after the child is born, to enforce him to pay his rightful cost. That means better ways of hunting down the father and getting child support, but also making the father pay far more in child support.
The problem with the above option is that it's pretty much impossible to force the father to provide emotional support for the child (or the mother, plenty of kids are given up for adoption or just abandoned). biologically and emotionally if a man refuses to provide emotionally for the child the women is going to be placed in a position of crying for the child or abandoning them to foster care; which in this culture is seen like a horrible thing (in truth I think a newborn placed up for adoption has a very positive life outcome, going to rich families that desperately want you helps to ensure your well taken care of).
Thus I think a good solution is to tackle the other end of the spectrum. Make men less available for reproduction. The best way to do this is a well known trope in sci-fi, a sterility plauge. Make it such that men are less fertile. Only a subset of men are fertile, and they can only reproduce so much (multiple sexual partners decreases sperm count so much as to render them infertile, limited mating is required to have any chance of reproduction). This approach helps to make fertile men a higher commodity. If all women in the world desperately want children, but only a few men are capable of assisting them, then women would start fighting for the desire to 'catch' a perminate mate to have children with.
Couple the above idea with government supporting children, to prevent population from decreasing, so that there was strong incentive for women to have kids and you could get into a position where women would fight for fertile men.
Go a step further, and make women the primary bread winners or people in power (easily done using the plague option as well, via making men less common to be born due to early miscarriage of male fetuses, or due to men being relegated to 'baby making' and not having time to focus as much on a job) and you get a society that would actually objectify males sexually.