Yep that should work. Now that you've seen how the clip connects to the anode you'll have an easier time in the future, I usually twist a wire around a flathead screwdriver and connect the other end to the ground strap around the back of the tube and then poke that under the cap, using it both to discharge the tube and to compress the clip to pop it out.
Take care around the neck of the tube too, the pins and their glass to metal seals are fragile, and there is a delicate glass exhaust pip in the middle. It's protected by a plastic cap on most tubes that size but that plastic can turn brittle and crack easily.
I normally advise against preemptive replacement of working capacitors, but if you're going to do it, make sure you use good quality parts and install them carefully, I like Panasonic, Nichicon and Rubycon in particular.