Sure you can.
The script might look something like this:
#!/bin/bash
# Set permissions
chmod 755 /home/user/Downloads/*
# Move files
mv /home/user/Downloads/* "/home/user/New Location/"
Save this as something like /home/user/download-script.sh and make sure to run chmod u+x /home/user/download-script.sh so it can be executed.
To have this executed periodically you could use cron. Enter crontab -e and create an entry like this:
0 * * * * /home/user/download-script.sh
This would lead to the script being executed every hour exactly (so, say 1:00 am, 2:00 am, etc.).
You can expand on this of course. Depending on whether there might be folders in you Downloads directory, you might want to add -R to the chmod-command to make changes recurse into those directories for example. Hope this helps :).
Note: You might want to be careful about when you use 755 as far as permissions go! Does everybody really need to execute those files? Maybe something like 764 is enough already.