<p>Cura has a layer view feature that lets you watch a simulation of the extruder head as it lays down material at each layer. Is it possible to get Cura to show a time stamp as it does this? That would let me set reminders to check a print just at certain critical times.</p>

These estimates tend to be very approximate, even if Cura has the accurate acceleration values for your firmware. An error of 100% is not unusual.
What you probably want is an alarm at a specific layer (a few before the critical ones). You might be able to add this to Octoprint fairly easily - it does support plugins which can provide (for example) pushbullet notifications.
I'm not sure that 'critical' points are much more likely to fail than other less predictable things (like bed adheesion failure, extruder jams, filament breaks) - unless you're testing features (and then hopefully you can print only a slice of the part).