"He is walking against the wall."
Translation:הוא הולך נגד הקיר.
That's interesting. Hadn't thought of that. The "wall"--not talking about the western wall but the other wall, the one to separate--is a matter of controversy, so much so that some people will not admit that there is a "wall." (Historically European cultures erected walls against Jewish people, sometimes referred to as Ghettos.) At first I assumed that the person was walking alongside the wall, but then one might expect perhaps ליד. My hunch is that it refers to a person walking opposite the wall as in on the other side of the street adjacent to the wall, but without context the sentence is obscure.
Or maybe it could mean something like "swimming against the stream or current" or even trying to accomplish something but the wall is an obstruction. But what I do not understand is, why does this sentence not need את? Wall is the direct object, unless נגד is one of those words that does not require את?
No, wall is not the direct object here. That can be seen from two things here. First, the verb הולך is an intransitive verb and never takes a direct object. Second, there is a preposition. Direct object never goes together with a preposition. When there is a preposition, it implies that it is an indirect object or a preposition phrase.
From commenting elsewhere, I can see you really struggle with את. But let me give you some pointers. First, if there is a preposition, any preposition, no את. That also means that when you have a verb, if you can ask the question "what?" and answer without adding any preposition, it will probably be direct object. I say probably, because the same verb in both Hebrew and English doesn't always follow the same pattern. But for the most part, yes.