Simple trick: you are a manager and probably spend > 50% of your time in meetings. Schedule yourself some meetings to spend time keeping up with tech.
I also found when I was a manager that there are some natural downtimes that occur when a lot of PM's or your peers are out, take advantage of those intervals since you'll have less meetings.
Also also, tech doesn't change that quickly. You can probably get to a level of understanding of some new tech by just reading up on it. This is just to understand if it is applicable to your area w/o having to go through the download/config/install/frustration loop.