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You can do either, but it looks like you have a good setup for putting in a garden- you could do some shrubs, or groundcover, or flowers, whichever you prefer. It's best to first determine what soil characteristics you have, such as drainage and pH, to see what plants will do well there:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-texture-jar-test.htm

Here are some tips for low-maintenance gardening if that's what you're looking for:
https://www.gardeningknowhow.com/garden-how-to/info/low-maintenance-landscaping.htm
https://www.gardeningknowhow.com/special/spaces/gardening-with-ease-the-low-maintenance-landscape.htm

Quick growing annual vines include cardinal vine and trailing nasturtium. Climbing roses are a classic look. Other possibilities include clematis, native honeysuckle vine, Carolina jessamine, climbing hydrangea and jasmine. American wisteria is more tame than the Chinese variety. Just make sure your arbor is big enough and sturdy enough to hold its mature size. Since I don't know your location, make sure your choice is appropriate for your location and the amount of direct sun the plant will receive. Lists often include trumpet vine, English ivy and wintercreeper (Euonymus fortunei) but these are invasive in many parts of the US and should be avoided, in my opinion.

That would really depend on your growing condition, amount of light, whether it be shade sun, and many other factors. Give us your zone information, whether the spot is sunny or shady, and any other information you can, and we will be glad to help.

Evergreen trees often make good privacy screens because of the denser foliage, and many can grow in pots. These articles will give you more ideas:

https://www.colostate.edu/Dept/CoopExt/4dmg/Trees/Shrubs/screen.htm
https://www.gardeningknowhow.com/special/spaces/plant-a-privacy-screen-plants-that-grow-fast-for-privacy.htm
https://www.gardeningknowhow.com/special/spaces/creative-screening-with-plants-good-borders-make-good-neighbors.htm

You may need something large such as 10-15 gallon containers, but I believe this can be done. Dolomitic lime, and wettable sulfur will be your bests friends with containers such as this. I would recommend Junipers, or other evergreens, as they will be better suited for containers, and the dense foliage would help drown out more sound.

Here is an article with more information: https://www.gardeningknowhow.com/ornamental/shrubs/shgen/evergreen-shrubs-for-privacy.htm

The Leyland cypress can tolerate a wide range of conditions, and will have a relatively low needle drop compared to most conifers. Here is an article on the trees: https://www.gardeningknowhow.com/ornamental/trees/leyland-cypress/growing-leyland-cypress-trees.htm