Script with root privileges running by an ordinary user
 

I need a script starting app server listening on port 80. This server can be started by certain users not having root privileges. How can I achieve that? I guess I can allow only execution of the script for other users, so they can’t open it and look inside. What should be inside?
I plan to put
sudo /b>appserver<

How can I specify root password in script?
Is there some other more appropriate to Linux solution?

You can configure this in your sudoers file.

Maybe you can try creating a new group, add these certain users to it and privilege this group with just execution rights, no read and write access.

Like wpn146 said: sudo.