Your question relies a lot on tenses. In addition to present, future, and past tense, there's also progressive and perfect tense. Progressive and perfect work in conjunction with present/future/past, to talk about the state of an action.
Progressive tense is an action that is currently ongoing and perfect tense refers to action that has finished and the time that this occurred relies on perfect, future, or past tense.
For example, progressive present reads like
The washing machine is running.
Currently, as we speak, the washing machine is running.
The washing machine was running.
In the past, the washing machine was running.
Your question refers to an ongoing action in the past (my dog has been running for an hour) which would be progressive past-- the tense shown in the last example.
The most natural way to explain your situation would be, "This action started a hour ago," because it fills the rules applying to the proper expression of progressive past tense. The other option of "This action got started a hour ago," (and I'm replacing your 'an' with an 'a,' because 'an' is only used if the word following it starts with a vowel) would still be technically correct but still awkward because the 'got' is unnecessary to the meaning of the sentence, and slows down its flow.
You don't need to use the got because the past tense was already established in your sentence by using "started."