bash run function with different user

Yes, this is possible

#!/bin/bash
function1(){
   echo `whoami` 
}
export -f function1
su username -c "bash -c function1"
exit 0

You can't do that, at least not directly. (But see Richard Fletcher's answer.)

Each process runs under a particular user account. By default, that's the same account as the process that invoked it. sudo lets a process running under one account launch another process that runs under a different account.

When you invoke a shell function, it doesn't launch a new process. With some modifications, your script should give you something like:

sudo: RunStefano: command not found

In the new process created by sudo, there is no RunStefano command; the function is local to the process running the script.

You need to isolate the function into a separate executable script; you can then invoke that script via sudo.

Incidentally, you also need to change the apostrophes around /usr/bin/whoami to backticks:

echo "Ciao, `/usr/bin/whoami`"

And you should read the documentation for the sudo command; it doesn't have a -c option.