I think the second sentence sounds nicer, but I was always taught that time should be at the beginning. So, what's right?
- Back in 2000 for example, I built a robot.
- For example, back in 2000, I built a robot.
You can use "Back in 2000" at the beginning of the sentence, at the end of the sentence, or any position inbetween.
Back in 2000, when Larry Smarr left his job as head of a celebrated supercomputer center in Illinois to start a new institute at the University of California, San Diego, and the University of California, Irvine, he rarely paid attention to his bathroom scale.
For yielding to the call of sticky goodness back in 2000, David's 11-year-old behind was bounced from his Douglas County elementary school.
I remember back in 2000, when Hillary Clinton first ran for the U.S. Senate, and this is what we found.
Could the state that gave us a murky verdict back in 2000 give us a clear winner this coming Tuesday?
Both your sentences are fine. As "for example" is a parenthetical, there should be a comma before and after it.
Back in 2000, for example, I built a robot.
OP's construction contains three elements...
A: Back in 2000
B: for example
C: I built a robot
...which it's perfectly grammatical to present in any sequence. The only semantic implications are that by default, for example modifies the immediately-preceding clause.
Thus in the sequence as given above, the primary focus of the example is that it was back in 2000, rather than the fact that I built a robot. That's to say, the speaker is calling attention to how long ago he was doing such things, rather presenting robot-building as an exemplar of his technical skills.
I'm assuming that (as is common in such contexts) the speaker isn't really presenting a randomly-chosen example with no particular significance. Most likely he wants to emphasise his level of skill, and/or the fact that he's had such skills for quite some time.
Idiomatically, in speech there's a tendency to get the less important parts of an utterance out first, because whatever's said last usually sticks in the mind of the audience more (plus those "less important" parts can simply serve to catch attention, as in throat-clearing, "Hmmm,...", "Well,...", etc.).
This can be established by noting that in speech, native speakers normally say...
John said he isn't coming.
...rather than...
He isn't coming, John said.
But in writing it's much more likely you'll see that rendered as...
"I'm not coming", said John.
That difference in the written form arises partly because writers habitually seek to capture the reader's interest by exploiting "slightly unusual" word order (hopefully, without detracting from fluency/readability).
TL;DR: All six sequence permutations for the three elements are grammatical, but in speech (not writing) there's a slight tendency to put the most important one last (thus, probably A or C). And there's a slight tendency to put the least important (B, or the least important of A/C) at the beginning of the utterance.