Finally is often used to introduce the last in a series of statements. When used in this way, finally is especially common in speeches, presentations, or in formal writing such as in an academic paper or a job application.
For example, a person might end a presentation by saying:
Finally, I’d like to thank the conference organizers for putting together this event.
Eventually means at some later time or in the end. It can be used to talk about something that will happen in the future. For example, you might hear a basketball fan say:
The Celtics have some good young players. In a few years, they’ll eventually win the championship.
Or, you might hear a gardener talk about a tree that will grow to a tall height, as in:
This tree will eventually reach a height of 50 meters.
Ultimately can mean: in the most important way or at the most basic level.
When it has this meaning, speakers often use it at the beginning or in the middle of a sentence, as in:
Ultimately, it is a question for the voters to decide.
It is ultimately a question for the voters to decide.