I'd split the napkin calculations into "passenger ICBM" and a more efficient scenario where the rocket just gets you to altitude, and the rest happens by skyhooks that recycle all their energy.

Ah shit, I'm doing this, aren't I? Stay tuned for the edit.

Edit:

Alright, to get vertically up, it's around 2 km/s delta v. It would make sense to launch from a jet in the stratosphere, Virgin Galactic-style, so we'll neglect drag, and assume this is all lost on re-entry. (Maybe you could use the heat for manufacturing somehow? Another time)

To get to an antipode suborbitally on your own, it looks like it's the same as for low-earth orbit, which is kind of obvious in hindsight. So, 8 km/s. Distances in between are going to be in between, and we'll measure by fuel used.

Now we pull out the rocket equation, and assume an (effective) exhaust velocity of 2 km/s, for a really pessimistic case, or 20 km/s for an optimistic scramjet-type case. Let's assume each passenger counts for 200kg of dry mass, including luggage and the craft itself.

(e^2km/s^ ^/^ ^2km/s^-1) * 200kg ~= 444kg propellant.

(e^8km/s^ ^/^ ^2km/s^-1) * 200kg ~= 10,700kg propellant.

(e^2km/s^ ^/^ ^20km/s^-1) * 200kg ~= 21kg propellant.

(e^8km/s^ ^/^ ^20km/s^-1) * 200kg ~= 98kg propellant.

Meanwhile, for a conventional airline, we'll assume 2L/100km per passenger to make room for modest future improvements in efficiency. For an antipodal trip, that's 400L * the density of jet fuel, so roughly 320kg.

Scramjet cases all win out, and the bad rocket case is in the same ballpark with skyhooks (but takes only a couple hours and you get to be weightless in space). The trick, of course, is that none of these run on pure jet fuel. Conventional rockets will probably will run on some hydrocarbon and some LOX, and a scramjet is very chemistry sensitive and will need something special like LH2. LOX looks like it comes at around 1USD/kg in bulk, while A1 is going for 0.81USD/kg at the moment, so that's not a huge issue. Hydrogen is easy enough to make, but LH2 is god-awful to handle to the point where even SpaceX doesn't want to bother.

There's also still a lot of unknowns about the expense of maintaining hypersonic airbreathing craft - it might well dwarf fuel costs in even the most advanced, refined case. I don't know how easy a few hundred kg per passenger is to achieve with conventional rockets going into near-orbital trajectories. There's also the issue of upper-atmosphere pollution, which could hurt the ozone layer.

I think the conventional + skyhooks case is most interesting scenario, looking at the issues and advantages. Virgin Galactic is taking a b-line for profitability and mass access, and there's startups working on the kevlar skyhook. Who knows, maybe this will be a thing starting in the 2040s?