OK, the two Blacks with no White in the switch box is really odd. But forget that. The point is that the new outlet you want to mount MUST have both Black (Hot) and White (Neutral) as well as bare Ground, and you need to find them. Where? Well VERY likely in the fluorescent light fixture. Or, more precisely, in an octagon box it should be wired into. At that point, based on your description, there is a feed of Black, White and Bare coming from the beaker box. Then there are splices so that the canopy switch (pull chain) in the fixture is wired to feed power to the fixture's ballast coil, AND there are connections to send power from the Black Hot source out to the wall switch, PLUS a return from that switch that also feeds to the ballast somehow. I agree that's a messy system and redundant.
One important wish you have is to NOT need to install a new box for this new outlet. You really would prefer to do that in the existing box with the wall switch, using the combo device you linked to. That MIGHT be possible if you have some way to install a new wire cable from the light fixture to the switch box. Your posts so far suggest that might not be easy. Then MAYBE there's another way, but we need more info. You used the term "romex" to describe the cable entering the switch box. By that, do you mean that the cable between the switch box and the light fixture is one of those spiral steel shells with loose wires threaded through it? Or, is is a common plastic-coated cable with all three wires tightly sealed into the outer flexible plastic sheath? For that matter, how many such cables are there between those two boxes - only one with 2 Blacks and one Bare in it, or two cables with only one Black and one Bare in each?
Where I'm going with this is that, IF there is one (or two) flexible steel shells carrying loose wires through it, you MAY be able to disconnect those (with all power OFF, of course!) and use them to pull a new set of wires though that shell. You could convert that cable to a "14/3" system: one each of Black, White, Red and Bare. Or, IF there are two such shells but no apparent White in either, you might be able to pull sets of Black / White / Bare through EACH of them to establish a White (Neutral) line to use. If you could do that, then Black brings to the switch boxes the Hot feed from the fixture's source from the breaker panel, White brings the companion Neutral, Bare brings the companion Ground, and Red takes the Hot output from the switch back up to the fixture to control the lamp.
If you can do that, I will also recommend removing the pull switch in the lamp fixture from the circuit and adjusting the wiring connections there so that only the wall switch controls power to that lamp.
If this idea cannot work, the alternative probably is to install an additional mounting box in the wall or ceiling - someplace handy and easy to feed a cable to - then feed it power from the source cable in the light fixture box and mount a standard duplex outlet in the new box.