A determiner isn't necessary here in English - "I'm going to grandma's" is also accepted and correct. The reason that it is possible to add 'my' here is that it's so much more common to speak that way in English than in Polish.
NB: This only applies to relations who can still be addressed using their relation-name ie. 'mum' - 'I told mum' is fine but 'brother' - 'I told brother' is not
"Idę" means "I go" so "I go to grandma's in an hour" should be right? First it marked as incorrect "I go to grandmother in one hour) and now also "I go to grandma's in an hour"
Duo should surely accept our "I go to grandma's in an hour" – absolutely identical to my answer, too – but Duo rejected this in favour of "I am going to grandma's in an hour" as its proposed "Correct" solution.
Could it be that Duo simply forgot to include go + grandma's in its list of acceptable answers?
I can see how "za godzinę" might imply "within an hour", but I'd prefer more lexical precision here.
- Within an hour = w ciągu (jednej) godziny, w przeciągu (jednej) godziny.
You can say po godzinie to refer to past events, indicating that an hour has passed when something happened. I believe it's the same in English.
Here's an example where it works in the present tense:
Po godzinie dzisiątej obowiązuje cisza nocna - After ten o'clock pm, it's rest time.
The only way this tense/aspect can be used for recurring actions, is if you want to express annoyment: He is constantly/always nagging!
I've looked through some of your other comments and I'm under the strong impression that you are not a native speaker of English. Even if you were, this matter is so obvious, it's not even up for debate.