Let's put aside for a moment the existence of "days" in space (which is disconnected from any geographical line on earth), what would be the Halacha for someone that will stay on the International Space Station, or on a hypothetical colony on the moon? Would it have nothing to do with the count of days in the place he DEPARTED from? Is there any alternative?
There are two separate realities involved with space travel:
The first is when one is in the lower atmosphere, and thus “within” the reality where day/night and dates are established by sun’s movement and position vis-à-vis Earth.
The second is when one is in outer space and thus “outside” that reality.
It’s a valid (and AFAIK, an open) question where exactly to draw that line. Many Poskim weighed in on it in association with Astronaut Ramon’s tragic mission on Columbia.
With regards to the first reality, I presume you are referring to multiple orbits around the world in a 24-hour period, (you don’t need spaceflight for that, it can be achieved in regular plane flight closer to the Poles,) which some might thing could bring Shavuos many days earlier. However, that has already been amply addressed in a correspondence between the Rebbe and Rabbi Schochet. The gist of it is that crossing the Dateline is not devoid of geography, and when you are “within” the world, you can’t create a “personal reality” that doesn’t exist anywhere in the world. Or said differently: There are only two days/dates in the world at any given time, and the only question is which one you are personally aligned with.
Additionally, the fact that many Zmanim-related Shaalos are created for one who moves quickly through timezones (such as on the ISS if we assume that its within the first reality) is no inherent license for “making up something that works”.
With regards to the second reality, how it is addressed is not necessarily going to have any influence on the first reality, as they are two different situations.
What are the rulings for someone that spends time in the Arctic or Antarctic circle (or better yet, at the north or south poll)?
That has been discussed here before.