16 by 8 Keyboard matrix
Easiest would be to use 2 arduino's, which are connected in some way.
One of them should tell the other one what to do. I suggest you make the one reading the buttons the "master" and the other one it's "slave" device.

You could use 8 inputs, for each column and 10 outputs for each row.
Then you check all 8 inputs per column (pretty rapidly) and then you'll be able to read every button.

You could choose to either send changes or just every value to the slave device.

The slave device would have to do something similiar, but with 2 outputs, to address the right LED's/segments.

Alternatively you could use shift-registers.

There actually are a lot of tutorials on this to be found on the web.
You should check out "Arduino multiplexing" or "Arduino led cube" tutorials.