Yeah, if you own a bike with such a headset (and that's nearly all bikes for about 50 years and still being made) that spanner and a good, solid large crescent wrench are valuable tools.

When you do this, I suggest you first back off the locknut entirely. (sometimes but not always you can back it off and slide it up the stem. Other times the stem must come out or at least be lifted some. And loosening the stem can be an issue, but if that's the case, addressing the stem now instead of 10 years and weather from now will be a blessing you will won't regret. (Grease the bejeesus out of the stem's quill before you put it back in!) With the locknut off or lifted, look to see if there is a "key" on the washer; ie a small tab that fits into a machined slot cut vertically into the steerer tube. If so 1) hurray! Headsets without that tab can be one of life's lesser curses with the headset not wanting to stay tight and 2) you need to take reasonable care not to twist the lockwasher so hard you distort or shear off the tab. So grease the threads and the top of the washer. Turn the headset cup only when the lockwasher is loose, then hold the spanner and handlebar in one hand so it cannot turn and tighten the locknut with the other wrench against both spanner and handlebars.

That looks like a good, solid headset what should be a joy to adjust. And if it is of lesser quality, I am sure a stock and easily available Tange headset will be an very straight forward replacement.