Best way to make connection with rubber contact pad
This is more difficult than you may have thought. The telephone circuit uses 48 V to power the telephone, dialing, and on/off-hook signalling and speech circuits. It also uses up to 100 V AC for ringing signal. They also get the odd whack of lightning induced current. You can't connect your micro directly to it.

Figure 1. DTMF tone matrix. The fourth column is defined but rarely implemented.
The keypad switches the DTMF (dual-tone multiple-frequency) tone generator chip. The buttons are generally arranged in a 3 x 4 grid which is scanned by powering the rows (or columns) in turn and reading back the columns (or rows). In your photo I can see the traces running to the chip. The significance of this for you is that neither pole of the contact switch is grounded so you can't use a simple pull-down transistor arrangement to do the switching. You'll have to use miniature relays or opto-isolators (taking care to get the polarity right). Meanwhile, see if you can get a part number from the chip, read the datasheet and learn from that.
The whole project might be a bad idea!

Best way to make connection with rubber contact pad?

You don't. You make contact with the PCB. You would need to attach wires to the chip pins. Each relay / opto would short out a row and a column.