A bacterium is in a glass. Every second it splits in half. Each of the resulting bacteria splits in half the following second, and so on. The glass is filled a minute later. In what time could the glass be filled if, initially, there were 4 bacteria in it?

As per your question, bacteria divides every minute. So this is clear that every minute their population will double itself. e.g If they fill half of the bottle in 5 minute, so in next minute (6th minute) the population will double and will fill the entire bottle.  Now getting to your question , if the cup gets completely full in 1 hour and bacteria is doubling every minute then its very clear that just a minute before the cup was half full. So in 59 minutes, the cup would be filled half.