My first thought is you need to clean your shifter. There is a ratcheting mechanism in there which is the thing that makes the distinct clicks when you shift. When the grease is old the little pawl (the thing that locks into the teeth of the ratchet wheel) will sometimes not want to rotate properly which can cause your shifter to skip gears.
Removing the shifter cover, spraying it with plenty of wd-40 or a similar product and shifting up and down a bunch of times should solve the problem in most cases. If that doesn't help you might have to clean it a bit better using for example cotton buds and a solvent or cotton buds sprayed with wd-40.
This phenomenon is called a 'sticky shifter' iirc. Here is a video on how to possibly solve it:
(Note: the issue shown in the video is not your exact issue but the video does show how the ratcheting mechanism works and following the steps in the video should probably solve your issue as well