Learn Spanish (Latin America): Por último, escribo el mensaje.

Both are common connector phrases that start with por. While por eso gives a consequence ("therefore" or "that's why"), por último orders a sequence, meaning "lastly" or "finally".

Could I use por fin instead of por último to mean "finally" here?

No, they have different meanings! Use por último for the last item in a list or sequence. Use por fin to mean "at last!", which usually shows relief that a long wait is over.

Why does this phrase use por instead of para?

Por último is a fixed phrase. Por is used in many common fixed expressions that act as adverbs or connectors, such as por favor (please), por supuesto (of course), and the current lesson's topic, por eso (therefore).

Why is it el mensaje and not la mensaje?

Words ending in -aje are almost always masculine in Spanish. Even though it doesn't end in the typical masculine -o, it takes the masculine el, just like el viaje (the trip) and el garaje (the garage).