The way I see it the two factors that can break you on a climb are the "leg strength barrier" - when you reach a point where your legs can't produce the larger force required to keep turning the pedals, or the "heart rate barrier" where you redline your heartrate and the muscles start to burn. There are tradeoffs to make either way, and different techniques work for different people - you'll have to try a little of everything and find out what feels most comfortable for you.

As previous posters have stated, seated climbing is more energy efficient overall. Assuming the same grade, gearing, and cadence, if you stand to climb your heartrate will jump higher than if you remain seated. However when you stand, all of your weight going onto the pedals makes it much easier to produce the force needed to push on the downstroke.

Personally I often take lesser-grade climbs seated (to conserve energy) but will stand up on something really steep (when I run out of gears). But that's just a generalization. If I hit a low-grade but fairly short roller, I might just stand up instead of gearing down, because I'll judge that I can crest the hill before my HR jumps too high and it's faster to push the bigger gear. Or, if the hill is mild enough to stay seated but is really long, I might prefer to alternate between standing & seating - I'll stand to give my legs a rest for a bit, then sit to let my HR drop again, and repeat. Or, if I really want to give my legs a hard workout, I might stay seated through a steep climb and just force myself to spin it out.

When I stand to climb because I want to (as opposed to when I stand because I've run out of gears) I typically shift up between 2-3 gears to get enough resistance so that the pedals drop more slowly and I can have a smoother motion. Reducing cadence when I stand also helps slow the rate at which my HR jumps.

If you ride the hill until you feel as if you won't make it, and then try to stand up, you won't be able to sustain it. When you already feel as if you won't make it, your heartrate is already in the redzone and you're too taxed to meet the added energy demands of standing up. If you want to try standing you have to do it before you burn out.

Standing also requires some timing & balance, so if you haven't tried it before, try it out on a flat or low grade hill until you feel comfortable with it. As you're pedaling in a comfortable gear, shift up 2-3 gears and stand. I prefer to keep my hands on the hoods for this; I feel like I have better balance with the wider grip, but YMMV.

Regardless of whether you are sitting or standing, try keeping your HR down by pedaling a little slower and dropping to a lower gear if there's too much resistance, until you run out of gears. And if you push til you've got nothing left and still can't make it, remember to unclip before all forward motion stops and you fall over (yes, I'm speaking from experience here.)

The nicest thing about riding the hills is that they make you strong, and fast. Don't let them make you nervous - they're good for you! Let us know how this upcoming ride goes, and good luck!