Cutting pipe with machine
I'm not sure what kind of machine you're thinking of, but I usually cut metal pipe with a pipe cutter like this:
Images courtesy of Lowes.com. No endorsement intended or implied.
In your case, the pipe cutter will be difficult because you need clearance all the way around the pipe, and it appears that your wall is concrete and/or brick and it would be a lot of work to make clearance to swing the pipe cutter all the way around the pipe.
In your situation, I'd suggest using a simple hacksaw where the blade extends beyond the frame, something like this:

It will take a while, but you'll get there in the end.
If you're looking for a power tool way to do it, you could use an oscillating cutoff tool or an angle grinder with a cutting disk. These are both going to get through the pipe much more quickly than you will by hand, but you may not be up for buying a (potentially) expensive power tool right now. Here in the States, you can buy cheap ones pretty inexpensively, use it for your intended purpose and be excited if it lasts for the next job, or you can buy a more expensive one that'll last most of a lifetime. They're often available for rent at tool rental stores.
For cutting plastic pipe it depends on the type of pipe. For hard/rigid PVC/CPVC/ABS type pipe, you use one of the above cutters†. For soft/flexible PEX type pipe, there are simple &quot;scissors&quot; style cutters, and fancier ratcheting style cutters (that simply increase the force with less hand effort) similar to this:

For capping plastic pipe, it again depends on the type of plastic pipe you're using, but your local hardware store (or their website) will have a variety of plugs you can choose from. Pick up one in the appropriate size along with the proper method of attaching it (glue, crimps, etc.)

† Yes, I'm aware that smaller diameters of CPVC piping is flexible, but it's not as flexible as PEX. It can probably be cut with the ratcheting scissor style cutters, but I've not tried it, so I can't recommend it.