That depends a bit on what additional assumptions we are making. If we're treating minds as immaterial and not necessarily determined by physical processes, there isn't much reason to assume you'd inevitably occupy every possible position.

If we're treating minds as a phenomenon that's determined by physics in some way, our current understanding would suggest that over an infinite timespan, every possible physical configuration would occur, including ones where you are a fan of FC Barcelona.

Of course, the notion of self gets confusing in these types of scenarios.

I think your answer is absolutely fantastic - simple, clear, touching the main aspect of this apparently silly question. I believe in free will (thus in an immaterial mind), but I am still amazed at how easily you treated this heavy burden of eternity. Do you suggest that in some cases our mind becomes irreversible (immaterial mind)?

Instead "Can I remain the same?" ask yourself, "can I remain sane?" in an eternal life.

Boredom will set in sooner or later. You will be so bored you can't fathom it without going insane. Even if all the material and intellectual and spiritual pleasures will constantly surround you and change around you, you will wish you'd have died and got out of the confines of knowing everything you can know, experiencing everything you can experience.

If I were to live an eternity, would it be inevitable that at some point I would be the opposite of what I am now? For example, I am a devoted fan of Real Madrid, FC Barcelona's historical enemy. Is it a certainty that at some point I will be the fan of the rival FC Barcelona team? Suppose the two teams and their rivalry will also exist forever. PS: In case we have an immaterial mind and we possess free will. — Eugen

The question instantiates what I think is a "misconception": at one point you speak as if the "I" is distinct from the beliefs it holds (If I were to live an eternity) and at another point you equate the "I" with the beliefs it holds ("I would be the opposite of what I am now") as if they are one and the same thing.

Good point! That is a part of the mystery - can one identify with something to a point where that thing becomes part of himself forever or things that we are strongly attached to and believe are part of us are simply going to fade away at one point under the heaviness of eternal time?