How should I properly connect outdoor string lights to power outlet?
It depends if you don't mind eventually (see below) drilling some of the brick's mortar in a few spots.
You'll need Wire Clips or Cable Clamps. You may also find the Copper version in the right Plumbing Aisle of a big box store to be most attractive and longest lived. If possible, paint your screws and washers (if any are used) or only use stainless steel screws and washers.
I presume you'll drape the extension cord over the gutter's downspout and run behind that to the outlet. If not, then you'll need more clips/clamps.
There are Adhesive Clips/Clamps (usually in the Seasonal Aisle) and I would start with those until you decide on final routing and looping. But fortunately, your outlet box accommodates a cord while being securely shut...so that's a great start.
And yes, I would and you can use the outlet box's screws for 2 clips/clamps...I would use both on one side. I'd suggest using the left plug and the right side's screws or the converse. To keep the extension cord fairly relaxed and happy in its quite low drop from the outlet box before the upturn toward the gutter.
If the cord is too stiff for the outlet box to close or to stay plugged-in. Then, using a bar or bracket from the Plumbing Aisle or Hardware Aisle with the 2 bottom screws may be needed for that taming.
And then, plan on coiling and mounting any extra extension cord to the eave's underside, for the cleanest look.