Could I get into a C.S. Ph.D program after an undergraduate from a totally unknown school?
The reputation of your previous program will be relevant when applying to PhD somewhere else. However, this by no chance eliminates your chances of getting into a good PhD program if your CV shows your competence compared to other applicants from other schools. In other words, in cases where the schools (and probably professors) are not well known, having publications in internationally reputable venues (not easy to do in undergrad) along with good standardized test scores might carry more weight than school-related attributes such as GPA and maybe even recommendations. (These will still matter, but less so when compared with GPAs and recommendations from highly known institutions/professors)

I will add this however. Being in a small and newly-established program can have other potential drawbacks. Assuming that a new and small program equates to less experienced professors and less budget, you will be missing out on the higher quality mentoring and availability of the required labs and equipment. Also, the student body will also be smaller, so you will have less chances to find people to collaborate with and share ideas. These drawbacks can be just as bad as the perceived reputation of the school for PhD admissions.