Could an A320 pilot fly an A321 without further training?
Legally, yes. Airlines may want a check ride to make sure the pilots are aware of the correct rotation angles—A321 tail-strikes around 12°.
If not, how much training would he need? Maybe he has to practice landing, because of the different length of the aircraft.
Depends on the airline, but probably that (in a simulator—you don't take a transport jet out just for training).
What big differences in the cockpit do those planes have?
A placard somewhere.
A321-100 will have different fuel system switches, because it's tanks are not split to inboard and outboard. A321-200, the one with winglets, should have the same wing—and fuel system—as the rest of the series.
Engine instruments show different values depending on what engines you have (RR and IAE ones show EPR, the others show N1), but that is just the symbols on the screen.
There may also still be some early aircraft around that still have CRT displays with slightly different graphics—the graphics was simplified with switch to LCDs.
That's about all differences there are.