You have an incorrect receptacle installed at your apartment. Some genius just replaced the existing 30A dryer receptacle with the incorrect 50A range receptacle for some reason rather than doing it right.
You only real option is to get the landlord/super/etc to have the receptacle replaced with the proper 30A dryer receptacle. The cord you have on the dryer is correct, albeit the old style 3-prong version.
First you need to figure out the parts you need. Turn the power to the circuit off and remove the outlet. You need to figure out the gauge of the existing wiring. Since it is a 30 amp circuit, the wire should be 10 gauge, so my answer will mostly assume that.
Now you can go to the store and buy a 20 amp outlet, a face plate, a 20 amp breaker, and a breaker blank cover. Make sure you buy a breaker that is compatible with your panel. Check the outlet to make sure it can handle the wire gauge you have. If you can't find a suitable outlet that can handle 10 gauge wire, then you'll need a couple small pieces of 12 gauge wire, or a short piece of nonmetallic sheathed cable that you've stripped, and some wire nuts.
Back home: Turn off your main breaker and remove the cover from your breaker panel. Disconnect the wires going in to the existing breaker and remove the breaker. Install the new breaker and connect the black wire to the breaker. Find the bus bar where all of the white wires are connected and connect the red wire there. Put the cover back on and use the breaker blank to cover the empty spot you created from removing the double breaker.
Back at the outlet: If your outlet can't take a 10 gauge wire, use the 12 gauge wire to create pigtails: cut a black wire to about 8 inches, strip about 1 1/4 inches of insulation off each end, and use a wire nut to attach it to the 10 gauge black wire. Repeat with white (attach to red) and ground. Connect the black wire to the HOT terminal on the outlet, the white/red wire to the NEUTRAL, and the ground to the ground screw. Then push it all in to the box and put the cover on.
Best Answer
No. That 120V plug can give small amounts of electricity, like to spin the drum or run the blower. Normally the heat comes from gas.
What you bought there is an electric dryer. That thing requires INSANE amounts of electricity, and the little 120V plug cannot possibly provide it.
You can search the room (or other potential places where a laundry room would have made sense) for a large socket the size of your fist. It either has 4 slots... or 3 slots and looks like the scary clown from Halloween. (And it does kill people).
If there is no such socket on the premises, and you don't own the place, talk to the person who does. If you own it (or have permission) then you will need to call in an electrician to install the large circuit breaker, wiring and socket. He will also change the cord/plug to 4-prong, because 3-prong is illegal because it's unsafe. This is serious work, and should not be attempted by a DIYer new to electricial work. Success will depend on
If none of this can work out, then back to the store it goes. They should not have sold it to you.
If you bought this thing used eg. On Craigslist, then you made the common mistake of not checking what power it takes. I regularly score very low prices on 3-phase shop tools, because the seller bought it without thinking about power requirements (because he didn't know anything about power) and I go "heh heh heh" as I cart it off. Often he had bought it from someone else in the same pickle.