Your first sentence doesn't mean what you want it to mean.
No one better than me would be able to take full advantage of this unique opportunity.
This means that while there might be people who are better than you, they wouldn't be able to take full advantage of the unique opportunity.
It has the same form as:
No one X than me would be able to Y.
It could be applied to something like this:
No one taller than me would be able to fit through this doorway.
While it can be parsed, it doesn't have the meaning that you want it to have.
Your suggested rephrasing, on the other hand, would have the desired meaning.
It has the form:
No one would be able to [X] [Y] than me.
It could be applied to something like this:
No one would be able to [drink this glass of water] [faster] than me.
So:
No one would be able to take full advantage of this unique opportunity better than me.
This means that you can take better advantage of the opportunity than anybody else could.
There are, of course, several different ways of expressing the same general idea (depending on the specific context):
I'm the best candidate for this business course.
I'm uniquely suited to take advantage of this opportunity.
This is a unique opportunity I can't afford to pass up.
What you should actually say is subjective and only you can know if it works or not.
But, between the two sentences you provided, it's the second one that actually means what you want it to mean.
