Solely has two meanings: it can be used as a synonym for "only" (which may also include "entirely"—see examples [OED]), or it can mean "as a single person"; without the aid or involvement of anyone else.
"The questions at issue do not relate solely to Egypt and the Soudan." (The questions don't only relate to these countries, but to others, too.)
"In all his acts he relied solely on himself." (He relied only/entirely on himself.)
In your example it means "I alone am responsible, and no one else". In this case it is not a synonym of "only" (it is closer to meaning "alone") as your second example means something different:
"I am only responsible for the project" (I'm responsible for the project, but not anything else.)
However, it can be synonymous if used in a different way:
"Only I am responsible for the project"
although it sounds somewhat stiff or awkward.
I don't think there is an easy "rule" to differentiate between these words, as they are closely related, and can mean different (or the same) things according to context and even word order.