Use <PERSON>... (I like "Stars & Stripes Forever" personally, but do ignore the rallentando going into the third repeat of the first strain!)
If you've any exposure to him you can probably hit 120 beats per minute to within +/- 10; if you've any musical training that's closer to +/-2 bpm. Let's assume +/- 6bpm, because I don't know you. That's an error of one part in twenty.
...to calibrate a pendulum's length...
Holding the top with one hand, occasionally giving it a kick at the bottom of its swing to keep it going—pushing it at the bottom will perturb its timing the least. Once you've got the length marked out of a 120bpm pendulum, go ahead and rest your arms a moment while gathering material for the next part. This solves your problem of not knowing the local acceleration due to gravity. (Which we hope is constant to better than 5% over a span of hours. If not, you've got bigger problems.)
...to calibrate a sundial.
Quadruple the length of your pendulum—doubling its period—and hang it from some improvised frame. Within seated arm's reach of your new grandfather clock erect your gnomon, mark your starting point, and start your grandfather clock. (Remember to give it its kick at the bottom—not like pushing kids on the swing.) Make a tick somewhere every 60 (or 100, if you're ambitious) ~seconds~ and after 60 (or 36) ticks mark the end of the hour at the shadow's tip. And do it again. And again. (Counting this many beats is do-able, but it takes discipline. Again, you'll be in good shape if you've some musical training. Particularly if you've been a brass player in an orchestra--plenty of hundred-plus measure rests in that repertoire!1)
Now build yourself a bevel gauge:
mark out eight times your three-hour count. There's your Earth-day. Now in 346 of those go to your gate location prepared for, at most, a nineteen-day wait. 312
(That is, assume your one-part-in-twenty error was in the direction of counting too slow: you want to show up 1/20 *365 days "early" so that even your slowest count gets you there a good half-day before your gate opens.)
This won't work if...
* Local gravity varies by more than a few percent over a few-hour span. By itself, I don't think that would wreak too much havoc on you, biologically. I'm guessing you could survive elevator-like levels of gravitational variance2 for a year. But I'd be seriously worried about whatever's causing that level of variation. Nearby orbiting attractor? Probably no atmosphere, then. Your system orbits an infalling pair of black holes? Good luck surviving the irradiation. &c.
* There's not enough light to use a sundial. For long periods. If it's overcast for a few presumed-days before you can do this work, not a problem: pad the wait-window by those few days. But if we're talking many many weeks of no appreciable visible light, then will this biome support your food needs for a year?
* The planet rotates slowly enough that you can't perceive the shadow's motion in mere hours. In this case you've got temperature differentials on the light and dark sides of the planet that are going to cause killer (literally!) weather. Not just a problem for you but, again, for your support biome. I don't doubt things could live in perpetual 200 mph winds, I just doubt you could digest them!
It's better than the non-answer "it can't be done" because now you're not tied to an area within (gate open-time)$\times$(running speed) of the gate location.
It's better than heartrate because that can be soooo variable. As a runner I know that my resting rate may be as low as the 40s, but on a stressful (Earth) day my resting rate might be nearing 60. +/- one part in five is no good. Even at the bad end of the tempo-range you'd be hitting one part in twelve.
It's better than sleep cycles because we're actually diphasic sleepers: in darkness we can easily sleep three or four hours then come wide awake, feeling refreshed as after a full night's sleep, only to re-tire an hour or two later. That's in our preferred environment, when well-fed. Throw hunger, stress, and unnatural stimuli into it and I don't think you're going to get one part in twenty accuracy.
1 - Much rarer, but better-equipped for this particular exercise, are change ringers. 801