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If someone were "off to work", it would mean they were at home (e.g.) and were just leaving for work: "I'm off to work; see you later."

If someone is "off work", it means they are absent from work. This could be for any reason, including vacation.

There is very little difference in meaning between "for a month" and "for one month". The indefinite article is often used to mean "one", and in most cases reads better.

It will take an hour to reach our destination.

I'm off to the UK for a week.

Using the number one convey a sense of accuracy, however:

It will take one hour to reach our destination (60 mins)

It will take an hour (or so) to reach our destination.

Difference between the tense usage is that "I have been informed" implies that the "informing" happened very recently and is still relevent to the present, whereas "I was informed" could have happened at any point in the past. The latter is not wrong, but the former is better.

It is even possible (though much rarer) to use the simple present in this context.

I'm told Robert will be off for a month. The action happened so recently that it is virtually in the present; the action is so relevent to the present that it has directly influenced all of your own actions since it occurred.