Is it possible to control a 12v led matrix with an arduino?
You can use ULN2803 driver IC on sink side (cathode), P channel MOSFET on anode/source side of common anode LED matrix. Since you need only 4 sinks you can pair every 2 i/o of ULN2803 so you can drive/sink 1A max load per row. This IC is cheap and widely available.

MOSFET current and speed should match those of ULN2803. You can use BJT instead of MOSFET, but voltage drop on BJT will be higher.

Similar design is here: 8x8 LED matrix