Learn Hardcore Arabic (Modern Standard): اليوم أكتب رسالة في الحافلة.

Yes. Arabic often begins with a time expression like اليوم (today) to “set the scene.” This is very common in both Modern Standard Arabic and formal writing. It’s an example of fronting the adverbial phrase for emphasis or clarity.
You could also say أكتب رسالةً اليوم في الحافلة; the meaning is similar, but starting with اليوم highlights today.

Why is the verb أكتب first after اليوم? Is this a verbal sentence?

After اليوم, the core clause begins with أكتب (I write / I am writing), so it’s a verbal sentence (جملة فعلية):

أكتب = فعل مضارع (present/imperfect verb)
The subject is built into the verb: أ- indicates I.

Does أكتب mean “I write” or “I am writing”?

It can mean both. The Arabic present tense (الفعل المضارع) often covers simple present and present progressive depending on context. With اليوم (today) and the situation في الحافلة (on the bus), English often prefers I’m writing.

Where is the word “I” in the sentence?