A wall gets you started. A screen gets you the picture you paid for. Both statements are true, so the real question is sequencing.
What a painted wall costs you: texture, even on a well-finished one, which softens 4K detail into something closer to Full HD; uneven reflectance across the surface, so bright scenes go slightly mottled; and no control over gain, which is the screen's ability to send light back toward the seats instead of scattering it everywhere. A proper screen is flat, uniform and correctly reflective, and that is a measurable difference, not a marketing one. It shows most in white uniformity and fine detail. Your projector is doing the work; the wall is quietly undoing part of it.
Ambient light changes the answer completely. In a room with light you cannot fully block, an ambient-light-rejecting screen does something no wall can imitate: it rejects light arriving from the ceiling and sides while returning the projector's light from the front. On a white wall the room's light lands on the image and lifts the blacks to grey. On an ALR screen most of it is thrown away. If your viewing room doubles as a living room with a window, that is not a luxury item, it is the item.
If budget is tight, my sequence: better projector first, because that is the part you cannot fix later; a smooth matte-white wall while you save, painted properly, not the builder's finish; then the screen. Doing it in that order also means you see exactly what the screen adds on the day it goes up, which is a better argument than anything I can write here.
The one exception is light you cannot control. There I would bring the ALR screen forward, even at the cost of a slightly less ambitious projector, because the screen decides whether the projector's brightness reaches your eyes or the ceiling's.
Send us a photo of the wall in daylight and one at night, plus the intended size, and we will say whether the wall is a reasonable stage one or a false economy in your room.
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