"Is this course sufficient to prepare myself for Hartshorne's Algebraic Geometry? Or do I need to study more chapters?"
You need to to study fewer chapters, the exact number being (to first order approximation) zero.
What I mean is that Hartshorne uses a very restricted number of results in commutative algebra:
Hilbert's Nullstellensatz, Krull's principal ideal theorem, characterization of factorial rings by principality of height one primes, finiteness of integral closure and a few other theorems.
These results are very important and very hard in the sense that no student having just learned the relevant definitions could find the proof himself.
However you can then take these grandiose theorems on faith, as black boxes, or check their proofs (maybe later, at your leisure) in Eisenbud or other books on commutative algebra, like Zariski-Samuel or Matsumura.
But there is absolutely no need to read the 355 pages comprising the first 14 chapters of Eisenbud: they are very interesting but studying them seriously is not at all required and would actually prevent you from learning genuine algebraic geometry for a very long time.
I would advise you to very thoroughly study the first few chapters of Atiyah-Macdonald in order to familiarize yourself with the basic concepts, browse through the rest of the book and simultaneously start Chapter 1 in Hartshorne.