EV Wall Charger with built-in metering?
In short, yes

There are at least two consumer-oriented EVSE (Electric Vehicle Supply Equipment) brands that include utility-grade consumption monitoring. PG&amp;E, my utility, actually had a pilot program (which I was a part of) for submetering EV charging. Instead of installing a separate meter, these reported the consumption over time to the utility directly so charging could be billed at a separate rate. 

I don't remember all of the options (and I don't want to make this a shopping list), but I have a ChargePoint Home. The app provides consumption data. Similarly, JuiceBox has a more portable solution, which only requires an appropriate receptacle (typically a NEMA 14-50). You'd need a J1772 to Tesla adapter in both cases, but the ChargePoint Home would be a more permanent solution vs. the JuiceBox which is inherently portable.

Is this worth it, probably not

As many others have said, good luck getting your apartment manaement to agree. They'll need to pull permits for a new 40-60A circuit, have the work done, and access the EVSE's data. Given that you don't exactly have much bargaining power, I doubt they'll want to negotiate at all.

Economically speaking, $50/mo. translates to about 400kWh at my utility's EV charging rate. This might be a slightly generous assumption if they allow you to charge at any time because the rate I assumed is only for off-peak hours. Peak hours are much more expensive. Without any Time-of-Use, it'd still work out to 250kWh with my utility.

Your mileage may vary (literally), but I expect you'll likely average 300Wh/mi consumption, which means 250kWh will get you ~833mi and 400kWh will get you ~1333mi. If you drive the average of 1000mi/mo. (which is only about 30mi/day...), then you'd be right on track to pay quite fairly for your charging anyway.