"New to computer" and "new to project" are incorrect. You could be new to computers, in which case you are new to all computers, or you could be new to the computer, meaning this specific computer you are working on. You could even be new to a computer, meaning that you are new to a specific computer, but you are not saying which one. That would be strange, but I don't see anything grammatically wrong with it.
In your first sentence, that is correct because the speaker is new to the specific project he or she is talking about, and may or may not be new to projects in general. It would be a little odd to say one is "new to projects" since most activities people do can be considered projects, but someone who does not have much experience with computers could definitely say they are "new to computers".
Edited to add
As the other answer pointed out, in the case of computers the gerund is also acceptable: "new to computing". But this is only true because a "computer" is "something that computes". A "project" is not "something that projects", and therefore "new to projecting" is incorrect. It might mean that you are new to speaking loudly to an audience, or new to using a projector to display images on a screen, but not that you are "new to working on projects", which would be the correct form.