Learn Arabic (Modern Standard): التقطنا هذه الصورة العفوية سنة سافرنا إلى مصر. - We took this spontaneous picture the year we traveled to Egypt.

The word سنة (year) is a temporal noun (adverb of time) that is directly annexed to the entire verbal clause سافرنا (we traveled) in an Idafa (genitive construction). In Arabic, certain nouns of time can take a whole sentence as their second term (mudaf ilayh) instead of just a single noun.

Why doesn't سنة have a preposition before it, like "في سنة" (in the year), and what is its case ending?

While you could say في سنةِ (in the year of), it is standard to use this temporal noun as a direct adverb of time without a preposition. Because it indicates when the action happened, سنة is in the accusative case (mansub) and ends with a fatha, pronounced as سنةَ.

Why use التقطنا for "we took" instead of the basic verb أخذنا?

The verb أخذ generally means to physically take hold of or acquire something. For taking photographs, the standard phrasing in Arabic is التقط صورة (literally to capture or catch a picture), using Form VIII of the root ل-ق-ط.

What vowel does مصر take at the end of this sentence, given it follows the preposition إلى?

It takes a fatha (مصرَ) instead of a kasra. This is because مصر (Egypt) is a diptote (a restricted noun, or ممنوع من الصرف). Most names of countries and cities are diptotes, meaning they do not take a kasra in the genitive case, using a fatha instead.