I just moved into a new apartment and in one of the bedrooms, there is a light switch that switches one of the outlets on and off. Unfortunately, this outlet is placed in the opposite side of the room where I want the lamp to be. Would it be easy/possible for me to do the work and somehow tie the switch to the particular outlet that I want? Or should I hire a professional?
This will likely require a change to the wiring. If you want to install an outlet on the opposite side of the room, that will likely require pulling new cable. Depending on the location of the room, what is above and below it, this may or may not require cutting holes in the walls and/or ceiling.
While this can be a DIY job, if you are unfimiliar with electrical wiring then it is not something you should take on your own and should call a professional.
There are other options out there like wireless switches, or you could go the easy route and just run an extension cord.
If it's an apartment, meaning you are renting, then this should be a request to the landlord, and they may or may not have it done for you.
I'd instead consider an alternative...namely the wireless switches they now sell. You plug it in-between your lamp and any outlet, and then you mount the switch (which is wireless) next to your regular wall switch with those 3M removable mounting strips.
As a bonus, you can take it with you when you move to your next apartment.
You can check and see if any other of the outlets in the room also turn on and off. When doing this, test the upper and lower outlets of each receptacle separately; a two-outlet receptacle can be "split" so that its top and bottom outlets are on different circuits, allowing the outlets to be supplied by different breakers (allowing higher overall loads) or allowing one outlet but not the other to be switched.
If you are sure that the switch only controls that one outlet and no other, then you're best off asking the landlord to make any requested changes, or as DA01 says buying wireless switch setups for the one you want to control. You don't own this space (unless by "apartment" you really mean "condominium"), and messing with the wiring without knowing what you're doing can be extremely dangerous to you and to others living in the same main building.
If you were going to do it yourself, first (after turning off applicable breakers), you would want to remove the switch from the line to the outlet that is currently switched; you do this very simply by removing all the black wires that are currently connected to the switch, and splicing them together with a wire nut, along with an extra black wire that connects back to the switch. This gives the formerly-switched outlet a continuous supply regardless of the switch setting, and still supplies the switch with power for the new outlet it will control.
Then, you want to remove the outlet that you want to be switched from its current electrical supply. This is done by removing all black wires connected to the outlet, and splicing and wire-nutting them together, to ensure that connections to other outlets "downstream" of this one in the circuit keep their power supply. If there's only one black wire, simply "cap" it with a small wire nut and a wrap of electrical tape. Do the same with the white wire(s) in the box, and with the bare wires (if there's more than one they should already be connected).
Now, you will need to connect the switch to the outlet you want to control it with; this requires running a Romex 14/2 (or 12/2 just to be safe) line from the switch box through the walls to the outlet box in question. This will require cutting or drilling holes in the walls and ceiling in order to feed the wire, and to drill holes in studs and plates that block your path, and constitutes a pretty serious DIY project. You'll need to know how to properly run Romex so that you or someone else doesn't get a shock some years later when they nail something into a live wire, or contact a metal stud that's been energized by wire running through it. This is why I suggest letting your landlord do this; you can get it VERY wrong and cause a huge liability for your landlord (and if he finds out you did it, for you too).
Once the wire has been run, hookup is pretty simple; simply connect the black wire on the switch side to the other end of the switch, join the white and bare wires to the white and bare bundles already in the box, and then in the outlet box, connect the black wire of your new run to a gold screw on the right side of the receptacle, the white wire to a silver screw on the left side, and the bare wire to a green or black screw on the top or bottom. DO NOT connect these new wires together with any existing wires in the box (with the possible exception of the bare wire ground). Reinstall the switch and outlet in their boxes but don't close up just yet. Turn on the breaker(s) and verify that the switch properly operates the outlet by plugging in something like a lamp (a plug tester is a good idea too) and flipping the switch.