Learn Hardcore Arabic (Modern Standard): أنا أنتظر دقيقة في محطة القطار، ثم أعود إلى البيت. - I wait a minute at the train station, then I return home.

No—Arabic verb conjugation already shows the subject, so you can usually drop أنا and simply say أنتظر دقيقةً في محطة القطار، ثم أعود إلى البيت.
أنا is included for emphasis/contrast (e.g., “I am waiting…”, not someone else) or for clarity in some contexts.

What tense is أنتظر? Is it present or “I am waiting”?

أنتظر is the imperfect (non-past) form, 1st person singular: I wait / I am waiting.
Modern Standard Arabic often uses the imperfect for both simple present and present continuous; context supplies the nuance.

How is أنتظر formed, and what does the initial أ- mean?

The initial أ- is the 1st-person marker in the imperfect:

أنتظر = I wait
تنتظر = you (m/f) wait
ينتظر = he waits
تنتظر = she waits

The verb itself is اِنتَظَرَ / يَنتَظِرُ (Form VIII). It’s related to the root ن ظ ر, with the meaning “to wait” in this form.

Why is it دقيقة without “one”? Does it mean “a minute” or “one minute”?

On its own, دقيقة can function like “a minute” in English in this kind of sentence (“wait a minute”).If you want to be explicit about “one minute,” you can say: