This phrase is from the 2nd to last panel in the lesson "A cafeteria conversation" currently page 17 in the classroom.
When it was first introduced, あまり was only to be used with negative endings, and while I supposed technically this phrase goes on to become negative it seems like it is more split into 2 pieces separated by "じゃ" and "あまり" is in the first part used to characterize liking something not very much (but no negative to be found).
My attempt at translating it is thus:
I (only) somewhat like meat, so I won't have it.
Is that accurate?