Does your DM like Wuxia?
Wuxia is "a genre of Chinese fiction concerning the adventures of martial artists in ancient China [and] is traditionally a form of fantasy literature". The abilities of monks in D&D have drawn on this literary and cinematic tradition since the earliest editions of the game.
One particular source of wuxia abilities is the practice of qinggong, which when "exaggerated in wuxia fiction [gives] martial artists...the ability to move swiftly and lightly at superhuman speed, and perform gravity-defying moves such as gliding on water surfaces, scaling high walls and mounting trees".
While in my limited experience of Wuxia I don't recall anyone running up the rain, there was a memorable scene in Tai Chi Master in which the antagonist is hurling javelins at superhuman speed and the protagonist is running on top of each javelin in flight in order to close the distance over uncrossable terrain to the antagonist.
So I certainly think that 'running up the rain' is within the spirit of the kind of abilities that a 5e monk gets at ninth level; I am comfortable with it being "rules as intended" (RAI). That being said, even for the explicit ability of running up vertical surfaces, there has to be a liquid there to run on - air doesn't count, and the ability does not give one the ability to fly, levitate, hover, or move vertically without a supporting surface.
In order to satisfy rules as written (RAW), you need to make an argument about running "across liquids". Rain itself is not a liquid, but it is composed of falling liquid drops surrounded by air. If the ability allows you to run across the surface of a still pond, a generous DM interpretation would allow you to run across* the individual upper surface of each liquid raindrop. However, you would still need to account for the fact that these individual raindrops, unlike a solid wall, are falling even as you are pushing off of them, in the same way that it might cost you more movement to run upcurrent on the surface of a swift river. Were I to allow such movement as a DM, I would say that running up rain counts as climbing where "Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling." A DM who wanted the allow this use of the ability but more strictly limit would be within RAW to allow you to ascend the rain, but as "climbing difficult terrain", such that if you had 45 feet of movement you would be permitted to ascend 15 feet on your turn.
They could also require you to make a Strength check to push hard enough or fast enough off the falling rain to reach the next drop or to not slip: "At the DM’s option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check."
Also to note is that this ability works only on your turn, such that as soon as you complete your movement, you begin to fall if you have run up off of the ground.
*@Medix2 "to move across a liquid" may invoke the meaning of across as from one side to another. But across can also mean 'from one thing to another in succession in order to cross something else.' Pitfall Harry crosses waterholes by moving across the heads of the crocodiles, even if he puts only one foot on each head. This use of 'across' could work with raindrops.