As far as I know the permission system in Linux is set up in such a way to prevent exactly what you are trying to accomplish.
I think the best you can do is to give your Linux user a custom unzip one-liner to run on the prompt:
unzip zip_name.zip && chmod +x script_name.sh
If there are multiple scripts that you need to give execute permission to, write a grant_perms.sh as follows:
#!/bin/bash
# file: grant_perms.sh
chmod +x script_1.sh
chmod +x script_2.sh
...
chmod +x script_n.sh
(You can put the scripts all on one line for chmod, but I found separate lines easier to work with in vim and with shell script commands.)
And now your unzip one-liner becomes:
unzip zip_name.zip && source grant_perms.sh
Note that since you are using source to run grant_perms.sh, it doesn't need execute permission