If Machine A is a Windows box, you can use Plink (part of PuTTY) with the -m parameter, and it will execute the local script on the remote server.

plink root@MachineB -m local_script.sh

If Machine A is a Unix-based system, you can use:

ssh root@MachineB 'bash -s' < local_script.sh

You shouldn't have to copy the script to the remote server to run it.