Is it possible to set execute only ?
Yes, try:
chmod 111 <script.sh>
chmod a+x,a-r,u+r <script.sh>. User will be able to execute, but will everyone else ?
Yes, a+x sets User (owner), Group and All others to execute.
For User (owner) only to execute only, try:
chmod 100 <script.sh>
I hope that helps.
Mike