First some comments on the words used:
1. You will be giving the blood. The doctor will be taking it. Be careful not to give the opposite impression.
2. A "regular" test is something which happens to a set schedule: every month, or every year, for example.
3. A "routine" test may also be a "regular" test, or it may be a one-off event; but the important thing about it is that it is not an emergency procedure.
4. "Fasting" may be used instead of "on an empty stomach".
5. If you are having a blood test, then the fact that you will be having blood taken is understood.
Taking all of the above into consideration, I would express the sentences above as, "Tomorrow, I will have to see a doctor for a routine fasting blood test".
Finally, I would assume that "a routine blood-work" is the same as "a routine blood test".