Connect(solder) a Arduino to some buttons
This is going to take a lot of speculation on my part. There are a couple of ways of wiring up a push button (e.g. passively pulled-up, passively pulled-down) and how you "bypass" it depends a lot on the specific wiring and push button type (e.g. normally-open, normally closed). The switch looks pretty common though - like this one. That is to say it probably operates like this (schematically):



Start by getting a multimeter and doing a continuity check among the pins of the push button. I would venture to guess that the four pins of the push button are actually two electrical pairs (i.e. {T1,T2} and {T3,T4} in the schematic above). Without pushing the button you will probably get the two ringing pairs, and with the button pressed you will probably get all four to ring out to one another.

Lets go with that. You want to use the Arduino to "push" the buttons. You might be able to get away with connecting the Arduino GND to one side of the push button and a DIG pin to the other side, then setting the pin to LOW to "push" the button, and HIGH to "release" the button. It's hard to say without measuring how the push button is wired in circuit. 

The most generic way to bypass a push-button and put it under electronic control by a microcontroller (like an Arduino), knowing nothing else about it, is (once you figure out which pins get shorted by the button press) to put a "pass gate" (aka transmission gate") in parallel with it and control it with two pins of the Arduino. 



A pass gate is a PMOS transistor and an NMOS transistor with their drain and source pins (respectively) wired together. You turn "on" the pass gate by supplying the PMOS gate with a LOW and the NMOS gate with a HIGH. You turn "off the pass gate (surprise) by supplying the PMOS gate with a HIGH and the NMOS gate with a LOW.

The more sophisticated device to use would be an Analog Switch IC that fits your specific needs for number of switches, poles, and throws. A pretty neat little through-hole one that will handle up to four independent switches would be TI's CD4016BE.