Accommodation close to Hong Kong International Airport

I would suggest taking the Airport Express to Hong Kong station and staying at the Ibis, which is quite nearby.

You can experience the high-end shopping in Central; traditional shops and restaurants in Sheung Wan; modern bars, restaurants and clubs in the Lan Kwai Fong area and the Mid Levels or if you are feeling a little more adventurous, then WanChai. If you are feeling particularly touristy then the funicular tram to The Peak is also nearby and we'll worth a visit, weather permitting.

You can then check in the next morning on Hong Kong station itself and have a more relaxed onward journey.

The free option (except for food) would be to sleep at the airport. There are free resting lounges that are quieter than the main area. Of course you're still in the open and there's no guarantee of finding a seat.

The next cheapest option would be to take a bus to somewhere in the city, probably Kowloon, and find a cheap hotel/hostel there. But don't expect miracles: if it's cheap, it's going to be a slum.

Given that you have little time but your budget doesn't appear to be extremely tight, I think you should aim for the next level: take the airport express train to Tsing Yi or Kowloon, and find a hotel near the train station. The airport express is relatively expensive (about USD20 return) but it is fast: 22 min journey time from Kowloon, 10 min interval (reliable) between trains. Get an Octopus card (Hong Kong transportation card with pre-loaded value) at the airport counter and pre-load it with the fare amount that you're going to need.

It does look like all the hotels close to the airport are very expensive. There are cheaper hotels in Tsing Yi, but they're not real close to the train station. The hotels close to Kowloon and Central stations are pretty expensive too, so you may want to change to another train and head further the Kowloon district. In addition, at Central or Kowloon station, you may be able to check in, possibly even drop your hold luggage when you arrive the day before; this depends on your company, on the day and time and possibly on your ticket type so check ahead.

Kowloon offers a large range of accommodation. Beware that if it's cheap, it's definitely a slum; if it's expensive, it's probably not a slum but do check reviews and book ahead. But once you've checked into your hotel you can get out and experience the “real” Hong Kong.

With your body clock only 2 hours from HK you have a fair bit of time. If you want to go cheap, grab the Airbus (about HKD 35) to Kowloon and stay at a relatively inexpensive hotel. Expect to pay around HKD700 for a reasonable place (depending on season etc. it was much more expensive a year or two ago), or less than that for a smelly and poorly ventilated dump such as a guesthouse at Chungking or nearby- but IME still safe if relatively unpleasant. English-only is not much of a problem in HK. You can take the Airbus back or the Airport Express train, which is faster (and less prone to delays due to traffic) but more expensive.

Take the Star Ferry across the harbour (cheap), walk around Causeway Bay shopping district (free), marvel at Central etc.