I was targeted, harassed, sexually assaulted and raped by an off duty Austin police officer. Can I sue APD?

Anybody can sue anybody for anything, but in order for you to prevail against the department, you will have to come up with some way in which its wrongful acions caused you damage.  For example, if you had previously reported your neightbor but the Department took no action because it was protecting one of its own, you could have a case.  Ditto if it issued a weapon to your neighbor which he used to assault you when the department knew or should have known that he was unstable or otherwisse unqualified to possess a weapon.  By itself, however, being attacked by an off-duty employee does not create liability on the part of the employer.  There are also sovereign immunity issues.  To the best of my knoweldge, all 50 states allow you to sue the state or various government entities, they place restrictions and procedural hurdles on these claims which don't apply to suits against private individuals.  I don't know the rules in Texas, but one common one is that you are requireed to file a notice of claim with the municipality within a realtively short time after your claim arises.  If you wish to file a claim against the dapartment, you should move quickly to check out the requirements you have to meet to commence suit, and should consult  a local attorney.