Actually the audio is wrong. To be grammatically correct, it should be: مُحامٍ مهم (muHámin muhim). The word محامي (lawyer) is a special kind of word because it ends with a vowel (í) ي. When such words come without the definite article (AL الـ), the last letter is omitted and replaced with Tanwin.
It is Mubtada' indeed. However, it is not always possible to put Dhamma (-u) or (-un) at the end of the word. Typically in Arabic grammar books, in such cases they say ضمة مقدّرة (Dhamma Muqaddarah) meaning an estimated Dhammah. Because it should be there but the word's ending blocks that Dhamma for showing (for various reasons like the case here).
For example: عيسى وَلَدٌ ذَكِيْ (eesá waladun ðakiy); Meaning: Eesá (Jesus) is a smart boy.
Eesa: Mubtada' (subject).
Waladun: Khabar (predicative).
ðakiy: Cifah (adjective to Waladun).
Here, عيسى is a name that ends in Alif-Maqcúrah ألف مقصورة, that is ى (don't mix this with ي Y). This name is mubtada' indeed, but we cannot put Dhamma there because it ends with (-á).
muHámin is the same, we can't put Dhamma at the end because the noun itself has undergone a surgery - one vowel is removed and replaced with Tanwin.
Apologies, I was reading it a bit wrong here but anyway the main concept here does not change. The sentence محام مهم is actually a phrase composed of Noun+adjective only and not a complete sentence with subject (mubtada') and predicative (khabar). However, the rule of Dhammah and all that does not change - it's just essential to correct the understanding of the sentence.