It is a little unusual to give dates in all-numerical form in such a message, but that is a style choice.
One does not use "the" (or any article) before a date given all in numbers. This form is used only when the day is given as an ordinal, such as "the fifth of March". If one says "March 5" or "5 March" no article is used.
I think it is confusing to give both an end-of-vacation date and a return date. What the caller really cares about is when you will return the call, or when you can again be reached by telephone. So give only the return date, for example:
- I will be on vacation from 03.01.2022, returning on 31.01.2022.
- I will be on vacation from 1 January 2022, returning on 31 January 2022.
It would be slightly less formal to say "I will be back in the office on" instead of "returning on" but either is correct, this is a style choice. So one might say:
I will be on vacation from 03.01.2022. I will be back in the office on 31.01.2022.
(another phrase such as "back at work" or "taking calls again" might be substituted depending on the circumstances and personal preference.)
However, there is nothing wrong with "returning on" and many people use it. It is perfectly natural.
One might also include something like "I will return messages starting on 01.02.2022" if messages will not be returned on the day one is back in the office.
One might further include "For urgent matters, call Pat Smith at extension 5897." or some other form of alternate contact statement.