yes, when we type or write typically we don't write the diacritics or Harakat on letters, as we know how to read (most of the time).
Anyway, the -un sound is called Tanwin (or Nunation). You can find it on Wikipedia for a bit of history about such "vowel". Anyway. you can think of it, for the time being, as being a marker of an indefinite noun; Somewhat like the (a/an) in English, but it is added at the end of the word instead. Tanwin comes in 3 flavors: -an, -in, and -un; All depends on the position of the word in the sentence (being nominative, accusative, etc). In nominative, (singular/normal) nouns end with (-u), or in case of being indefinite, that's (-un).
From the three type of Tanwins, only the -an requires an additional Alif when written (but sometimes it is not written for orthographic convention, that's another story).
Here are some examples:
- A Syrian girl: بِنْتٌ سورِيَّةٌ (bintun súriyyatun) - Nominative.
- He knows a Syrian girl: هُوَ يَعْرِفُ بِنْتًا سورِيَّةً (huwa ya3rifu bintan súriyyatan) - Accusative.
- She goes to a Syrian girl: هِيَ تَذْهَبُ إلى بِنْتٍ سورِيَةٍ (hiya taTHhabu 2ilá bintin súriyyatin) - Prepositional (approx. terminology).
Typically, the last vowel in the last word at the end of the sentence is dropped when speaking, so all the (-tun, -tan, -tin) above are dropped and simply replaced with (-h), but here I've typed the full form to show how Tanwin takes place. You might need to enlarge the font to see the proper sign for Tanwin in the text above.