The dryer's instruction book will have instructions on how to change the cord. The most important thing is that right now (with 3-prong), there is a "ground strap" connecting neutral and ground on the dryer. This must be removed.
I think I see it right there; a metal foil that lands on the neutral terminal and connects to chassis. You must remove that. Feel free to use that screw for the ground.
If you have a volt-ohm meter, you can sanity-check this by measuring resistance between neutral and ground pins on the recep. It should be high resistance (0L or infinity ohms means high resistance).
It sounds like you're already onboard with this, but you must change the cord on the dryer to a 4-prong cord. It is illegal to continue with a 3-prong connection or to defeat ground in some way.
Old 3-prong connections are dangerous. If there is any trouble with the neutral wire, the dryer chassis will be energized with voltage. (meanwhile the washer right next to it will still be grounded, so touch both and whammo!)