Your alternative sentence uses the future form "be going" plus infinitive: "am (not) going to eat". This form is used to talk about future plans and predictions, so this version would be appropriate when the lunch-less week has not yet started. Because it is "this week" this sentence would make sense on the morning of the first day of the week (before you skip lunch for the first time), for example to explain to someone why you would not have lunch with them that day.
The original version of the sentence uses the present continuous: "am (not) eating". This tense is occasionally used to describe future plans, in which case it could be used in the same circumstances as your alternative. However, as the name suggests, it is also frequently used for an action that is happening now, so you could also use this sentence when your week of skipping lunch has already begun—anywhere from after lunch on the first day of the week to lunchtime on the last day of the week. (You wouldn't use it after the last lunchtime of the week, because then the action would be completed and you would need some past tense form, such as "I want to lose weight, so this week I skipped lunch".)
In summary, the original sentence can be used at any time from the morning of the first day of the week until lunchtime on the last day of the week, but your alternative sentence works best only before lunch on the first day of the week.