Connecting Arduino to a Monster iEZClick Remote
Both are possible, just the RF one is harder than the other. Without knowing the specific remote, it is common for remotes to have simple active low or active high buttons. When the button is pressed, the remote's microcontroller pin, as an input is connected to ground (or vcc), the remote sees the change, and acts appropriately. Your arduino would have it's gpio pins as (High Impedance) Inputs, connected to the microcontoller side of the remote's buttons.

If you need to control the buttons from your arduino, you would need a transistor or relay in parallel with the remote's button. Fairly simple.

RF is harder, as you would need to know the frequency and modulation, possibly the encoding if it uses any, as well as the communication protocol. You might luck out if it uses a standard rf transmitter/receiver IC, but it's not a beginner's project. At a bare minimum, you would need to understand radio frequency and have an oscilloscope or two. Possible yes, practical, no.