Neisseria gonorrhoeae is a fastidious bacterium, so your choices are not ample, but you might be able to get it to grow in nutrient broth or rich nutrient peptone broth supplemented with 10% sheep or horse blood. See attached publication for more details.
Selective media such as Thayer-Martin or Martin-Lewis agar would be better for swabs specimens. I also once used GC medium (by Difco) which you could use to produce chocolate agar using 5 percent sheep blood and you can make it selective by adding antibiotics like vancomycin, colistin and nystatin
I would send out a request for some BHI broth from your surrounding laboratories. It is such a common medium some one would be able send you some. I would grow the organism in broth and then remove the spent medium by centrifugation. Then re-suspend the pellet in fresh sterile BHI broth with 15 to 20% glycerol. Aliquot the suspension to 0.6 ml into 2ml freezer test tubes. Freeze them at -80C freezer. After a week or so thaw a tube to determine the CFUs on a BHI agar plates. As these fastidious organism, the titer still will not be stable, so every time you use it re-titer so that you how stable the organism under these conditions. All the best
I tried a bunch of freezing conditions in parallel (from the same Fastidious broth liquid culture batch, not shaken, but incubated at 5 % CO2) and 10 % glycerol or 10 % DMSO proved to be the best (higher percentage of glycerol didn't work for me). I do not get a lot of colonies when I revive from this stock, but it seems to preserve them reliably in liquid nitrogen. I don't know about longterm survival, but so far I always get the same number of colonies.