Is it possible for bacteria to enter through an injection in the upper arm then migrate down the arm and manifest there?

Yes: In fact bacteria can enter through your arm by way of an injection and manifest itself in your heart, people who use intravenous drug through infected needle often develop a type of infection inside their heart ( heart valve) called endocarditis, some people may also develop phlebitis (veins infection).

It depends....: If "injection" refers to a medical injection like an antibiotic, vaccination, cytokine inhibitor (like Humira®), and so on, the answer is no, or exceedingly rarely. If an un-hygienic self injection for illicit substance use, probably yes. Medical IV lines can become infected with spreading infection, but unlikely at these particular anatomic sites.