If I'm in Chile, Santiago, what is the best time of the year to see the star Alpha Centauri at the beginning of the night?
Alpha Centauri is circumpolar as observed from Santiago, so it is always well placed for observation. However it is best placed when it is on the meridian at midnight above the south celestial pole, which occurs on April 27.
The best time to observe an object is when it is on the meridian - the line through the sky that goes from due North to due South. The best time of day is when it's darkest, so I searched around for when alpha Centauri is on the meridian at midnight from Santiago. The day is April 27/28. So, it would best be visible at midnight on that day.
If you're a month earlier, then it will be on the meridian two hours later (2AM); a month later and it will be on the meridian two hours earlier (10PM). About now, it's crossing the meridian around 10:45 AM.
Alpha Centauri from Santiago is a circumpolar star, though, from Santiago. So technically it's in the sky all the time. Even today (mid-November), it's visible at night, but it's very low in the southern sky. It "crosses" the meridian at about 10:50 PM these days, but that's going from west to east ('cause it's circumpolar) and so low in the sky. It's high on the meridian at 10:50 AM, but the sun's out.
So again, while it's technically visible year-round at any time of day from Santiago, it's best viewed when it's on the meridian, and it's best-best viewed when it's high in the sky. Those coincide with night in late April.