The Prophet (S) cared about his appearance and used to fix himself up before each communal prayer. Although he lived simply, he gave due attention to personal grooming. Here are some narrations about how he approached that: https://www.al-islam.org/sunan-nabi-allamah-sayyid-muhammad-husayn-tabatabai/chapter-3-cleanliness-and-rules-beautification

The Qur'an also says:  “Put on your adornment on every occasion of prayer” (7:31).

I don't think it's correct to say that Imam Ali (A) didn't care about how he looked; rather he chose to live simply and did not own much clothing. One can still look after grooming, washing one's clothes, etc, even if one's clothing is old. Even on the day of Ashura it is said that Imam Husayn and his companions made the effort to remove unwanted body hair. (Sorry if that's sensitive but it's related!) Usually when people don't care about their appearance at all, it is a sign they are not feeling well, depressed, stressed, etc.  

However, it is good not to throw away clothing just because it is damaged. If it is possible to repair it, it is better to repair it and continue using it. After all, the Qur'an and hadith strongly discourage wasting, and the throwing away of discared clothing is becoming a big environmental problem today, especially with manmade fabrics such as polyester.

There is a virtue to doing things yourself, such as fixing your own clothing, but there is also no harm in taking clothing to a tailor to be fixed, especially since it gives the tailor income.

Basically you should use your wisdom and common sense. If wearing patched clothes will harm you at work or in other places, then it is good to wear clothing that is suitable for that environment.

Also, hadith encourage men to dress nicely in front of their wives at home, and wives to dress nicely in front of their husbands at home; it is good to try not to be an eyesore at home!

If you are representing Islam publicly, it is also good to have a neat and respectable appearance because people judge by appearance these days. Once, I remember I went to an interfaith program where the only Muslims were me and a Wahhabi shaykh with a stained, torn dishdasha that was well above the ankles. Guess which one of us ended up answering most of the questions. So it is good to be thoughtful of these things and to try to come across in a respectful manner, while at the same time not being extravagant.