If you are simply referring to the passage of time (and things taking a long time), then use "finally." I would use "finally" in your first example. I feel that "finally" also recgonizes the achievement of a goal.
If you are talking about a long (especially a long and difficult) process or journey, then "eventually" is probably a better choice.
It's good to meet you finally! -- Only finally works.
However:
Peter and Amy corresponded via letters for a year and half. They knew they would eventually meet in person, but they were not in a rush. Finally, the did meet, and already in love, married almost immediately and raised a large family of prolific letter writers.
Peter and Amy corresponded via letters for years and finally met when Amy attended a conference in Peter's city. However, they had a serious disagreement about politics, and eventually their relationship ended.