It appears to me that awareness of what time it is, awareness of how long a task has taken, and ability to guess in advance how long a task will take are distinctly separate skills - it seems unwise to me to assume that progress at one of them will translate to increased ability with another.
This is based on the observation that I'm abysmally bad at the first of those (I've been known to gain and lose entire days, even when I'm paying attention in preparation for an upcoming event) and non-awesome at the second, but startlingly good at the last if I rely on the appropriate non-conscious estimate-generator. If I spontaneously say "I bet I can finish writing that piece of code in 20 minutes", based on my track record it would actually be surprising for it to take less than 15 or more than 25, but ask me to take a shower that lasts no longer than 20 minutes and there's something like a 50/50 chance I'll be in there for 45.
Recent brainstorming sessions at SIAI (with participants including Anna, Carl, Jasen, Divia, Will, Amy Willey, and Andrew Critch) have started to produce lists of rationality skills that we could potentially try to teach (at Rationality Boot Camp, at Less Wrong meetups, or similar venues). We've also been trying to break those skills down to the 5-second level (step 2) and come up with ideas for exercises that might teach them (step 3) although we haven't actually composed those exercises yet (step 4, where the actual work takes place).
The bulk of this post will mainly go into the comments, which I'll try to keep to the following format: A top-level comment is a major or minor skill to teach; upvote this comment if you think this skill should get priority in teaching. Sub-level comments describe 5-second subskills that go into this skill, and then third-level comments are ideas for exercises which could potentially train that 5-second skill. If anyone actually went to the work of composing a specific exercise people could run through, that would go to the fourth-level of commenting, I guess. For some major practicable arts with a known standard learning format like "Improv" or "Acting", I'll put the exercise at the top and guesses at which skills it might teach below. (And any plain old replies can go at any level.)
I probably won't be able to get to all of what we brainstormed today, so here's a PNG of the Freemind map that I generated during our session.