Let's say someone is a Historian specialized in Middle-Eastern contemporary history, would they have opportunities to travel to Middle-eastern countries as part of their research? How common is this?
If so, would this be when one is studying to become a historian (let's say during Phd studies) or after?
Let's say someone is a Historian specialized in Middle-Eastern contemporary history
Sure, let's stipulate :)
Would they have opportunities to travel to Middle-eastern countries as part of their research? How common is this?
Do you mean opportunities paid for by someone else? It depends, but for an established historian or one who works for an established institution (either academic or a think tank) the answer would "Yes" and "very common."
If so, would this be when one is studying to become a historian (let's say during Phd studies) or after?
It depends. In contemporary Middle East studies there are lots of ways that historians, either students or graduates, might get paid to travel to or live in the Middle East. Off the top of my head:
In terms of expense, it depends on where you are but it's not usually all that expensive to travel to the Middle East. From America you're talking maybe $800-1000 per roundtrip flight, but from Europe, on Qatar, Turkish, Fly Dubai, etc. you're looking at more like $250. Cost of living in the Middle East is low.
If you want a good example of why historians travel to the region, (albeit not about contemporary history), I really enjoyed this recent piece in the New Yorker: https://www.newyorker.com/culture/culture-desk/a-new-history-of-arabia-written-in-stone
The bigger challenge for contemporary historians isn't the cost of travel but the number of countries or areas which are now de-facto inaccessible. Large (even overwhelming) portions of Syria, Yemen, and Libya are entirely inaccessible due to the conflicts in those countries. Saudi Arabia, while safe, doesn't issue tourist visas, so you need to be a sufficiently established business traveler to have a local guarantor, or get a student visa of some kind.
The exact answer to your question depends on a variety of factors, but in the vast majority of cases the answer is yes. In order to earn a PhD, students are expected to produce high-quality original research of monograph length. Such original research very often involves working with primary sources located in archives around the world. Naturally, for Middle Eastern history, many of those archives are located in the Middle East itself, and thus the historian must travel there in order to complete his or her research (and if not archives, they often are relying on some other form of primary source available only in-country).
But there are a handful of reasons why this might not be necessary. Archival materials are increasingly being digitized, making it easier to conduct research from one's desk without the need to travel (not to mention all the other kinds of sources available via the internet). Or the historian might be relying on a body of primary sources physically located outside the Middle East. Or maybe the primary sources aren't archival in nature at all. It is possible to produce Middle Eastern history without traveling there for research purposes, but comparatively uncommon to do so.
Now, it's a bit unclear to me whether you're asking specifically about the research phase of the PhD or about the entire educational process of becoming a historian. If the latter, then I should add that there are additional reasons for students to visit the Middle East beyond just research. Above all else is language training. Students of Middle Eastern history must learn at least one and usually several Middle Eastern languages and that normally involves some time spent in-country. It's theoretically possible that someone could learn the required languages without going there and then come up with a dissertation project that also wouldn't involve travel to the region, but that would be very strange, barring exceptional circumstances.