I watched the news on television about my former (old?) next door neighbor
[FORMER IS THE RIGHT WORD BUT YOU COULD ALSO USE "PREVIOUS" BECAUSE "OLD" COULD RELATE TO HIS AGE, SO "OLD" IS NOT THE BEST CHOICE]
WHO was arrested this month.
Now, two cops who come from the A neighboring city
[NOT "THE" BECAUSE THE READER DOES NOT KNOW WHICH CITY IS "THE" NEIGHBORING CITY, SO WE USE "A" TO MEAN 'ANY NEIGHBORING CITY BUT WE DON'T KNOW WHICH CITY]
have been investigating(?)
[YES, INVESTIGATING IS EXACTLY THE CORRECT VERB TO USE AND YOU COULD SAY "HAVE BEEN INVESTIGATING OR ARE INVESTIGATING]
about
[DELETE "ABOUT" BECAUSE THE VERB INVESTIGATE IS FOLLOWED BY SOMEONE OR SOMETHING, FOR EXAMPLE: THE BOSS / THE COMPANY]
my new next door neighbor for A different case.