Here'a a quick display of the data:
If you're prepared to assume that within each Gender/Var combination the counts are Poisson with the same mean, then this is reasonably straightforward.
The variance of a sum of independent Poissons is Poisson, so you can form a confidence interval for the expected sum within each Var/Gender category using methods you say you're already familiar with. Since the mean is the sum divided by a constant, you simply divide the endpoints of your confidence interval for the expected sum by that constant. Done!
However, there's a fairly clear suggestion of more variation within groups than the Poisson model would indicate ("overdispersion").
Some possibilities - you could look at fitting an overdispersed Poisson model, or you could look at a negative binomial model, and then in either case obtain a confidence interval for each subgroup mean from that.
For example, here's the fitted means $\pm 1$ standard error of the mean for a main effects overdispersed Poisson model with log link (done in R):
I'd expect similar results from a negative binomial model. (You would possibly prefer a model with an interaction term though the results would look fairly similar to these.)
These standard errors would be about 1.8 times as big as you'd get by assuming a straight Poisson model.