Advice replacing 1930s receptacle
Besides copper, the only other material suitable for conductors is aluminum.  In 1930 aluminum was a precious metal, as the Hall/Héroult refining process wasn't industrialized yet.  Far too precious to waste on wire, which was easier done by cheap, plentiful copper.  (Now it's the other way 'round lol).
This wiring looks like it may have been retrofit.  If so, it may be aluminum.  Scrape it with a knife to see if it comes up brassy or silvery.
That 2-prong receptacle is perfectly new, possibly 1980's more likely 21st century, noting the upward offset to leave room for an absent ground.  Inside the receptacle the metal bits are identical to the grounded type.
You can retrofit ground wires
That looks like armored cable, not conduit.  If you can freely pull out and replace the wires, it's conduit.  Otherwise it's metal jacketed cable.  Fixed metal conduit is allowable as a grounding path.  That metal cable may be, depending on type; AC cable was demoted to where it's  no longer allowable as a ground path.  No matter, you can retrofit just a ground wire, those rules were liberalized in 2014.
You don't need to follow the original cable, but given the ease of access you have, that's exactly what I'd do.  Do the same thing as the cable is doing, hop from junction box to junction box.  Those steel junction boxes are exactly what I use today for that sort of work, and they don't really age unless they're exposed to wet.
The layout of this wiring really did you a favor.  The splices are being made in the junction boxes, so the receptacle only takes 1 wire per side.  That's a knee-saver!  Just FYI, The 2 screws per side are there to optionally be used in one of two modes: a) as a handy-dandy way to splice 2 wires while also connecting them to the receptacle, or b) if you break off the tab, a way to control each socket separately.  Neither applies here.
If the box is grounded, you can ground the receptacle with a &quot;ground pigtail&quot;*. Search the back of the junction box for a hole that's a slightly different size than the others; that would be threaded for a #10-32 ground screw (note -32).  If absent, either use a grounding clip or drill and tap your own #10-32 hole (minimum size #6, thread pitch -32 or finer to bite enough metal to conduct electricity, don't use a sheetmetal screw!).
You cannot ground the receptacle via the mounting screws, but if the metal yoke bottoms out, clean metal yoke against clean box flange, that is acceptable. Typically the box is recessed shy of the drywall and that is not possible.
The ground path needs to go all the way back to the panel, but your retrofit wire only needs to reach a point with good ground that is served out of that same service panel,  e.g. say if you have a water heater that is grounded.
Aluminum wire
If you believe you have aluminum wire or other old wire you just don't trust**, a 'silver bullet solution' now exists in the form of AFCI circuit breakers.  They listen to the wire for any evidence of arcing, which is how most wiring problems show up.  Then, they trip to shut off power.   The concept of AFCI pretty much requires doing that in the circuit breaker; they make AFCI receptacles but they don't protect the wires feeding them, which defeats the purpose!   If you are installing AFCI breakers, they also make AFCI/GFCI combo breakers for a few dollars more. A GFCI breaker protects every point in the circuit, however, GFCI breakers are sensitive to wiring defects such as crossed neutrals.  Those are defects you want to fix anyway, but a GFCI breaker will trip until you do.
Long term anti-aluminum plan:
For small branch circuits (#8 and smaller) only use copper wire, because inspectors and homebuyers with small minds will freak if you use aluminum. Aluminum is perfectly acceptable for 50A+ feeders in #4 and above cable.  If I just spoke greek, get a book.
The wiring I see in your picture would be easily replaced if aluminum; so do that.  If you have aluminum wiring which is infeasible to replace, the problem with aluminum happened because they attached it to terminals that were rated for copper only.  Copper terminals don't like aluminum, zinc-plated aluminum terminals play nice with everybody.  So change to CO-ALR rated recepatcles and switches, and use Alumiconns instead of wire nuts for splicing. The purple wire nuts are no good. Alumiconns are actually miniature lug connectors, and like most lugs, they are made of aluminum.

* most grounds must be pigtailed anyway, note there is only 1 ground screw, that is because you are not allowed to use the &quot;handy-dandy use the receptacle as a splice point&quot; technique with grounds, they must be pigtailed so removing the receptacle does not break other grounds downstream.  That same rule also applies to neutrals in MWBCs, which is way above your skill level right now.  Upshot is if you see a neutral pigtailed &quot;for no reason&quot;, leave it that way.
** or new wire you just don't trust, namely backstab connections.  Backstabs are all the rage because they shave time when building new houses. However they are unreliable and are the #1 source of arcing problems. As a direct result, AFCI breakers are now mandatory in new construction.