I'm not sure "Being away" is a present participle or a gerund ? If it's a gerund , according to the rule "possessive before gerund" ,then "I'm used to his being away" should be correct. Am i right ?
Generally speaking either cases may be correct:
1. I saw him doing the job.
2. Despite his being away I did it.
In our case the gerund (the noun form of the verb in its progressive form) is more appropriate.
As a result, using a verb as a noun, it becomes possessive. The phrase should read: "I’m used to his being away”.