Prius Headunit

Maybe take a look at Crutchfield

Crutchfield: Car Stereo, Speakers, Home Theater, Pro Audio, 4k TV

They have a tool which shows what stereos will fit in a particular car and which give you steering wheel control. I'm not sure if the same stereo models are available over there, but it might give you a start?

 

When I went to their website just now a bunch of items popped up for 2016 Prius Liftback. Maybe email is the best way to go, though. They also have a live chat popup.

 

トヨタ プリウス | 室内・インテリア | カーナビ・オーディオ | トヨタ自動車WEBサイト lists the correct head unit options for a JDM Prius.

Note that only the first one is actually Prius-specific, the others are just generic Toyota units shared with other models. Running Google Translate, it looks like the steering wheel switches might be supported on all of those.

That said, I think you'll want something other than any of those, because I believe Japan uses a different FM broadcast band from the rest of the world.

Also, quickly looking at things, it looks like Singapore uses similar FM broadcasting technology to Europe?

 

My first thought would be to get a Toyota head unit for a European Prius. Then I realized (1) it might be outrageously expensive and (2) there might be unforeseen complications.

My second thought: isn't your Toyota dealer in Singapore eager to sell you some sort of head unit that would work? Or auto stereo shops in Singapore?

 

OK, next thought: so why not just do everything through your smartphone? And get a holder to clip it to the dashboard (Koomus has good ones, their magnetic ones are especially convenient). That's what I do anyway. I do GPS navigation, stream music, etc through my iPhone and use the Entune system just to handle the audio into and out of my iPhone. You'd probably need a Bluetooth earpiece.

 

good option.

 

A smartphone can't do everything a headunit can. No CD, AM radio, Energy Monitor, or backup camera display. Plus, you don't get steering wheel controls.

 

And vice versa. Thanks to apps, a smartphone can do much MORE than the head unit can.

For the things you mentioned:
CD: I download the tracks I like off my CD's into iTunes anyway, and my iTunes files are synced to my iPhone. I don't need CD's in the car. Lately I just buy music on iTunes or Amazon.
Radio: I stream music, I don't listen to AM or FM over the airwaves. I had to do this at home because my condo is in a dead spot reached by only one FM station. You can use one of the services like Spotify, Pandora, or Apple Radio. But I simply do a google search for radio stations playing the type of music I want to hear. For example, "classical music radio stations" (classical is not my thing, BTW, but can be soothing on a long trip). A bazillion links pop up from all over the world, most of which have an online streaming feed (some are online stations only, some broadcast and put their broadcast feed online as well). Far better than listening to local stations over the airwaves! I bluetooth that to my car stereo system. Now I think about it, you could get TV stations the same way if you want news? And just listen to the audio, not look at the video. I have an unlimited free data plan from Sprint for $50/month ($60 after all the fees/taxes); this might not be a good solution for those with limited free data.
Energy monitor: I never even thought of using it in my radio, forgot that it was possible. I think most of the info is available on the dash displays? I doubt if the OP could get a non-Toyota head unit that would do this?
Backup camera: Are standalone backup cameras with their own screen still available? Given that his Prius came with NO head unit, would it even have a backup camera installed?
No steering wheel controls: The OP wanted this and some third party head units will do this. With the iPhone option, I find the only thing I use is the volume control knob, which is as easy to reach as the steering wheel control, which I never use. I use one of these to keep the iPhone handy so I can use its touchscreen rather than the radio's touchscreen:
Pro CD-Slot - M | Magnetic Cradle-less Smartphone Car Mount

My take on this is that Entune is designed to seize control of my iPhone and limit my options to operations that are "safe" to do while driving. Unfortunately rough edges arise in this process, plus it forces me to learn yet another way to perform the same task - for example, two versions of Siri apparently. I prefer to keep things simple and to exercise self-control to not do things on the iPhone that I should not be doing in the car.

Getting back to the OP, this probably won't give him what he wants. And he would at least need some sort of head unit to power the speakers and handle bluetooth.

 

If the OP has an iPhone, maybe the Pioneer SPH-DA120 (the AppRadio 4) would be the way to go?

It has a backup camera input and a steering wheel control input, and there's a European model available, and supports CarPlay (Apple's official "put phone apps on the car display" tech).

Problem is that it doesn't support Android Auto. (Just its own proprietary AppRadio mode for Android.)