How can you determine the date using nothing but a star chart from the day and the time? I'm doing a science olympiad event called Reach for the Stars, and one question that I've encountered provides a star chart and time of day, and you must figure out what month of the year it is from. The strategy that I developed includes figuring out which direction is west (usually a star chart from the night and the Sun sets in the west), then going in that direction estimating which zodiac constellation in Sun is in (and in turn approximately what month it is).
However, I realize that this method would not work if it was taken late at night, much after the Sun sets, because although I am fairly sure that each hour the sky rotates about 15 degrees, I can't estimate what constellation the Sun would be in.
Is there a more definite way to figure this out?
Edit: The time in the example chart is 8:30 PM in the time zone of the observer
Example star chart:
First determine the North (South) direction by drawing a line from the center through Polaris (Sigma Octantis). The other principal directions follow from that.
You need to obtain the coordinates of a well-recognized star in equatorial coordinates (right ascension and declination), not too near the celestial pole. Given your chart, it's no problem as it shows the sky for a location near or even exactly at the equator.
As it's a question from a competition I will only sketch the path briefly:  The stellar coordinates allows you to calculate the local hour angle of it and from that the sidereal time. You then have to find the date where the sidereal time matches your given mean solar time. Depending on accuracy and possibly further details given in the task, you will also have to consider the equation of time.
OLD ANSWER where the time was not yet given:
A star chart is not unique, it's ambiguous for date vs. time. Just as you state: it could be from 22h today. Or it could be the sky at 21h in 14 days or 20h in a month. Generally every half month you have to remove one hour from the clock and it is identically valid.
Obviously a star chart is of limited use for times where the Sun is over the horizon or for the early twilight time where you still don't see any stars.
So what you can do is figure out for what date / time combinations the star chart is valid. In year books usually several of those are given for the star charts printed. Assuming that people usually observe in the early evening hours, one can get an estimate for the intended month though (stating the observation time).
Thus practically I'd use a programme like Stellarium and try find a month where for the chosen observation hour the night sky looks like shown. Mind also it is WAY more accurate to find North (South) direction in a star chart by drawing a line from the centre through Polaris (Sigma Octantis). All other directions follow.
You can break ambiguity of time and date, if your chart shows solar system objects like planets which change their position with respect to the stars.