Egyptian Arabic is dominant in a similar way to American English, i.e. Egypt had the largest movie industry and population, and everyone else could understand Egyptians while Egyptians had no need to understand other dialects.
What is interesting is the impact of satellite television. These new stations have global audiences and content that Arabs want to watch as opposed to their national censored channels. But their solution to pan-Arab broadcasting is classical Arabic, not Egyptian Arabic, and so there is a resurgence of the use of classical Arabic outside of courts and classrooms.
However, you will have difficulty understanding the dialect of the person you are speaking to. And if you have a dialect that you should understand, I would say it should be Gulf Arabic (Saudi Arabia, Kuwait, Qatar, Oman, UAE and of course my own Bahrain).
Economic (thriving free markets, not just oil-funded governments) and military activity in the Arab world is concentrated in the gulf, so unless you are trying to be a tourist (Egypt), the real value is in Gulf Arabic. Note that most Arabic teachers are Egyptian or from the Levant, and will try and persuade you otherwise ;)
1. Lots of people speak Mandarin at home. No one speaks MSA at home except us poor language students.
2. The set of languages binned under 'Chinese' is much more diverse than the Arabic dialect continuum.
3. The writing system is shared between several languages. You can write Cantonese (kind of) in hanzi and write notes that Mandarin speakers can read even though the languages are totally mutually unintelligble.
4. There is no dominant dialect of Arabic, like there is a dominant variety of Chinese. Instead every dialect shares the same formal register (MSA) which is taught in school like a foreign language.