That would be "na basenie".

It would be "nad basenem"

I'd say that it's "na basenie".

I guess it can work, added now.

In this particular usage they are very similar. With small things I would go with „obok”: szklanka stoi obok lampy (the glass is located near the lamp), with buildings, places: boisko jest przy szkole/obok szkoły (a pitch is next to schol). „Przy” with things can mean different thing, like „usiądź przy biurku” means „take your place at the desk” (this one specific place, not in front of it, not on the left/right of it). „Usiądź obok biurka” means "sit somewhere around the desk”. Also there are some common expressions that use „przy”, like "by the way" is translated as: „przy okazji”. And if I want to say that my work includes sitting in front of a computer (eg. something is utilized/produced during the work) I say: „pracuję przy komputerze”. „Pracuję przy produkcji piwa” - My job is directly connected with to beer production.

Aha, „obok” and „przy” go with different cases: przy basenie (locative), obok basenu (genitive).