Is it possible to say "I have ended up going"? or just should say "I end up going"?

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Sometimes, English speakers use the present tense when telling a story about the past.
So, "I end up going" is the historical present tense of the verb phrase 'end up.' It's a case of using the present tense to speak about the past.

"I ended up going" is natural, and a common way to relate historical events.
"I have ended up going" is not used.

I ended up going to the hospital. "Ended up" means 'finally' or 'after a long series of events' If you say "I went to the hospital" it suggests a quick trip, no thrilling misadventures along the way. A very boring story -- unless the fun begins at the hospital.