The IPA for "man" in American English is mən. Arabic, however, doesn't have the vowel sound ə. So, what's the most related sound in Arabic? I know there are some short vowels (Harakat) but I don't think they include this specific sound. In addition, when Arabic speakers want to mimic this sound, they presumably sound odd a little bit. So, I was thinking to "approximate" this sound to one of the existing sounds in Arabic.
First, American English "man" is [mæn], though I suspect that in some dialests it's really [man] (a lower version of [æ]). Second, a number of dialects of Arabic have [ə] (or close variants such as [ɨ]), though Classical Arabic doesn't. The Arabic vowel typically transcribed as "a" (fatḥah) is the IPA front vowel [a] which is pretty close to [æ].