I am making an 8x8x8 LED cube.
Which contains total 72 pins. 8 for layer select and 64 for selecting LED.
I have Arduino UNO. Can I do it with UNO?
If yes then which mux I should use?
Hi and welcome.
Use 8 x tpic6c595 (daisy-chained and connected to the Uno's SPI pins) for the cathode columns and 8 x p-channel logic-level mosfet connected to either 8 Uno outputs, or, easier, a ninth tpic chip for the anode layers.
Cue Bob!
You will need 64 current limiting resistors for the cathode columns, 8 x 10K pullup resistors for the mosfet gates, 9 x 0.1uF bypass caps for each chip and some big caps for smoothing the power, eg. 1000uF.
Paul
Welcome to the forum,
A couple weeks ago I'd released a library for controlling a LED cube driven by 8 bit shift registers, this also includes the 8x8x8 version. This topic includes schematics for a 5x5x5 LED cube, if you're interested you could take a look at it 
TOPIC HERE
Greetings,
Mike