You could say it either way :) "I really hope she is successful" "I really hope she succeeds" Those both refer to the future. "To be successful" can also describe person in the present who has done very well at something (for example, a man is a successful actor or a woman is a successful businesswoman). So if you were only trying to describe the person in the present, you would only say "This man is successful". You couldn't say "This man succeeds" (sounds strange).

The phrase should be "I really hope she’s successful.” The apostrophe S (here) is the contraction of “is,” and we cannot omit it. We could say “I hope she succeeds,” and the meaning would usually be about the same. “Succeeds,” after “hope,” refers to the future, whereas “is successful” could refer to present or future. Note that the 3rd person singular ends in S. Note that we don't use the subjunctive after "hope" in English.

Succeed is a verb and usually it contains a preposition. such as in, to ,by I really hope she succeeds in her new job. I really hope she succeeds, is correct.