Powering up LED strip - How many amps?

Usually strips have not individual LEDs, but are composed of a lot of "pieces" in parallel, and each piece is composed by three leds in series along with a resistor.

Usually the strips are composed by 60 leds/m (i.e. pieces of 5cm), but sometimes also 120 leds/m or 30 leds/m. Let's assume it is the 60 leds/m version (you can check the length of the piece you can cut - which should be 5 cm - and that it contains 3 leds).

This means that in 5m there are 100 pieces. Now, usually the 3528 leds absorb 20mA of current for each segment (the 5050 version should be rated at 60). This means that the whole led strip should absorb 2A @ 12V.

Now the power supply. 2A is the nominal current the whole strips should absorb. Add about 200mA for the nano, and increase a bit the current requirement not to stress the power supply: you get a 12V 3A power supply at least (note that, while 12V should remain the same, you can take any current greater than this value - if for instance you have a 12V 5A power supply it is perfectly fine).

Now, the nano. It surely can't handle the 2A current. This means that you can power the arduino nano from the 12V. BUT beware: you should connect the 12V power to the VIN input (so the onboard regulator lowers it to 5V). Then you can't power the LEDs directly from the nano, but you need a transistor to decouple them. You can take any nMOS with a drain current greater than 3A or an NPN with collector current >3A and build the appropriate curcuit to drive them, then connect the + terminal of the strip to the +12V wire, the transistor source (or emitter) to the ground wire and the gate (base) should be driven by the nano.

1) Is it important on how many amps, my powers supply will feed?

Yes. You need the PSU to be able to supply as much current as you will need for everything you are powering form it.

You don't say how many LEDs will be on this strip (only the length of the strip) and I am not sure how much current/power a single one consumes, but for illustration let's imagine there are 100 LEDs on your strip and each one takes 0.8W.

100 LEDs x 0.8W per LED = 80W.

P = VI, where P=Power, V=voltage, I=current.

So I = P/V = 80 / 12 = 6.66A -- call it 7A.

The amount taken by the Nano will be small in comparison - if you said 0.5A you'd be safe.

So with these figures you'd need a 7.5A PSU.

Considering that i need a particular value of amps to power up my LED strip, can the nano handle this many amps?

You can power the Arduino from the same power supply, yes. The Arduino will draw whatever current it needs. (Think of it this way: mains sockets are 13A - a desk lamp won't draw 13A.) What matters is that the supply can supply at least enough current, and that the voltage is correct.

As mentioned in the comments, you can't drive 3528s directly from the Nano - but you don't ask about that so presume you either know how to do it, or will ask another question.