Yes, a network meta-analysis is quite general and can be applied to any probability model including Poisson. The underlying assumption is that multiple interventions have been applied to different samples from the same patient population and we are interested in making indirect comparisons between the interventions (comparing effects across samples) on a population parameter such as the mean.
It might be challenging to work out the number needed to treat from only a rate ratio. This ratio tells you the disparity between two means, but there are any number of means that would result in that ratio. You may need to also know at least one of the incident rates to work out number needed to treat.