Though I wasn’t *particularly drawn* to fashion, I *admired good taste on a relatively low budget.

The only grammar issue is that that's an uncountable use of "taste", so it's inappropriate to use "a." The bigger issues are that it's not clear whose taste you're complementing and "attracted" and "built on" are unusual word choices for the subject matter. If I understand your intended meaning correctly, I would suggest: Though I didn't normally care about fashion, I noticed that she had put together a tasteful ensemble without spending too much.

"Though I wasn't attracted to fashion, I appreciated good taste on a (relatively low) budget." 'Good taste' in this sense is uncountable. 'Fashion' and 'good taste' are often opposites, so no problem here. To do something 'on a budget' is self-explanatory, no need for the qualifier 'relatively low'.

The first unnatural thing is: "a good taste". If I lick a chocolate ice cream cone, I will have "a good taste". That's what a "taste" is. A person can also display "a good taste". For example, "That he picked Bob as a friend shows his good taste in character". It would be difficult but not impossible to "build a good taste". But you could: "Jim went to art museums every day for five years. He built a good taste for art appreciation". Let's try this: "Though I wasn't attracted to fashion, I appreciated a low-budget product that clearly displayed good taste." This is a big improvement but it is still weak. Here are the problems I still see: (1) the word "though" implies that there is some sort of tension between the first clause and the second clause. But there is none. There is no contradiction between being attracted to fashion and appreciating good results. (2) The sentence is boring: "I appreciated good taste". Who cares what I appreciate? Doesn't everyone appreciate good taste? Why bother to say something so boring? Next let's try this: "Though I wasn't at first attracted to fashion, the good taste evident in this low-budget product overcame my indifference." Now there is tension. Now there is drama. Now something happens. Now the subject and the verb are in the center of the action: "GOOD TASTE OVERCAME MY INDIFFERENCE". That's exciting. It could even handle an exclamation mark!

Though I wasn't (directly) attracted to (a person's sense of) fashion, I (could) (definitely) appreciate(d seeing) (someone having/with) good taste built on a low budget. That's bsc all your options there I would say "I appreciate it when people have a genuine sense of fashion in stead of just grabbing whatever brand is big at that moment" or something along those lines.