The problem is almost definitely with the tube itself. If it were a driver IC problem, or insufficient high voltage, more than one tube would likely be affected. Your tube looks like it either has cathode poisoning or it's in the very early stages of losing its fill gas (neon). Losing gas is fatal. Cathode poisoning can be repaired if your clock has some sort of "tube healing" mode you can turn on to "bake out" the tube. depends on what clock you have.
Replacement tubes are still easy to find on eBay. Most listings call them IN-14 or NH-14 rather than NI-14. US$10 to 20 is a fair price for a single tube; cheaper if bought in larger lots. Look for tubes with long wires (proof the tube is not used) and avoid ones with short wires.
You say you know nothing about soldering, so I would say the replacement would be difficult. Removing the tube will be tricky because there are 13 wires to desolder. You have the choice of desoldering all 13 perfectly so the tube can be pulled out intact, or cutting the wires off flush with the tube's glass envelope and then removing the wires from the board one by one. Either way there is some risk of damaging the board with heat.
Having said that... with proper tools tube removal isn't too bad. I've successfully removed IN-14s using a Hakko 808 solder removal tool. With some skill and luck it can be done using either solder wick or a cheap "solder sucker" but that's a bit riskier.
Look into it later when the dust is clearing off the crater.
There is a burn mode, but it is to be used with caution. It can heal problems like this, but shouldn't really be used for extended periods or for anything other than correcting acute problems. Don't get into the habit of fixing things that aren't broken - it will only shorten the life of the tube if used without cause.
To get into "burn" mode:
Hold down the button for 10 seconds and then let go. "0" on the first digit should light up. Then you can cycle through all the digits using a short press until you get to the one you want to treat. Then leave it on that digit until the problem goes away. Don't leave the clock unattended while doing this and power off immediately if anything starts to smell.
You can come out of heal mode by pressing the button again for > 2 seconds,
While it looks like cathode poisoning, but it might be tube outgassing. The first one can be fixed, the second one will only get worse with time.
If you do need to change the tube, I would definitely advise you to snip all the wires close to the glass tube body and remove one by one. It's still tricky, and the hardest part without the proper tools is to get the holes clean enough to put a new tube in, even if you do get the old wires out. It's really a job for the correct tools.