Yuzhno-Sakhalinsk (UUS) to Sapporo-Chitose (CTS):

• 1h30 flight time


• 2 hours time difference


• 
  1.33 ratio.

You actually arrive before you left, without any of the international date line trickery:

(from Google Flights)

I'm pretty sure there must be lots of short 30-minutes-or-so hops across a timezone boundary which would result in a ratio of 2, so I'm not sure the ratio is really the best basis for comparison, though.

Edit

It turns out it's not as easy as I thought to find short flights crossing timezone boundaries, but I found one:

• Mariehamn (MHQ) - Stockholm (ARN)


• 35 minutes flight time


• 1 hour time difference


• 1.71 ratio

There's also Gibraltar (GIB) - Tangier (TNG), same characteristics.

Also Warsaw-Minsk, 2h hours time difference, 1h10 flight, same 1.71 ratio.

Note that the answers vary according to the time of the year, as:

• some areas switch to daylight savings time in the summer, but not all, and not necessarily at the same time


• some flights are seasonal

Additional edit

So this one is not a regularly scheduled flight, but it’s a commercial flight nonetheless: Bering Air flight 8E120 on 09/07/17:

• Nome, Alaska (OME) to Anadyr, Russia (DYR)


• Departure: 12:44 UTC-8


• Arrival: 10:31 UTC+12 (the next day, but that’s due to crossing the date line)


• Time difference: 4 hours


• Flight duration: 1h47


• Ratio: 2.24


• They arrived 2h13 before they left
  

I hope I got the DST and time zone math right :-)

You can also go forward in time with good results, see Almaty to Urumqi:

45 minutes flight which propels you 3:45 forward for a ratio of 5!

Maybe there is some trickery with flight time since the reverse flight is timed 2:50. Still they're 900 km away which should not translate to more than 1:30 in air.

For short flights, Vladivostok to Changchun gains as much as 1 hour depending on the airline:

Ural Airlines 881 Friday, May 17, 2019
Leave Vladivostok (VVO) 9:10am UTC+10
Arrive Changchun (CGQ) 8:10am UTC+8 (elapsed time 1h, ratio 2)

Going domestic here. Perm-Kazan leaves at 22:45 and arrives at 21:55 same day.

• 1h10 flight time (in reality less than hour spent in the air)


• 2 hours difference


• 1.7 ratio.

For a relatively short flight spanning a multi-hour time difference, it may be hard to beat Kusuluk–Reykjavik:

These two airports currently have a three-hour time difference, and the flight is 1h50m, for a ratio of 1.636. However, Greenland will go onto daylight savings time tomorrow, reducing this ratio substantially.

Long-haul flights can't compete on ratio, but on absolute time-gain for a commercially available long-haul flight, I nominate Reykjavik-to-Anchorage, which gains 50 minutes. (It would gain 110 minutes if it flew in the winter, but it doesn't.)

IcelandAir 679 Sunday, May 12, 2019
Leave Reykjavik (KEF) 5:10pm UTC+0
Arrive Anchorage (ANC) 4:20pm UTC-8 (elapsed time 7h 10m)

The time difference is closely related to how many lines of longitude you cross. So you're looking for as many crossed as possible, in the shortest time.

The time of a flight is closely related to its distance (for any given groundspeed), and is reduced if the prevailing air movement is from behind it (because airspeed is related to groundspeed plus speed of air movement/currents).

So you want to cross as many lines of longitude as possible, in the shortest possible distance, with the wind coming from behind. That immediately suggests you want to be travelling as close as possible to the north or south poles, and in a west-to-east direction for northern extremes, or east-to-west for southern extremes.

You are limited because too close to the poles,you lose time zones - nobody bothers with time zones being 12 hours different if you and a friend both start at the north pole and walk a mile in opposite directions. But the Arctic is almost certainly better for this than the Antarctic, because many countries are closer, with more interest in asserting territory (hence time zones), and longer explored/less isolated.

Your air flights at high latitudes are likely to be from ad hoc runways. You wont find big towns there. But even small communities may have local runways, and your question doesn't exclude those.

Putting this all together, I think your answer won't be found by looking at major airline schedules.

You need to find where in the arctic you could find the most northerly territories which have time zones. Then look for the most northernly settlements in those areas. Then, with a map centred on the true or magnetic north pole (whichever longitude originates at), figure out the shortest distance between settlements as far around the "clock face" as you can.

You'll have to do this manually, because there are unlikely to be scheduled flights, but that's how you'd do it. I suspect your answer would involve Arctic airbases or exploratory stations belonging to Russia and the US.

If you flew the path you found, you'd get maximum time zones per flying hour.

Clearly there are multiple short flights that will meet this criteria. In fact, I can easily get a ratio of over 100:1 by flying my drone across a timezone boundary. (30 seconds to do, 60 minute difference, ratio of 120:1!)

Once you exclude the flight of less than a few hours things get closer to parity, however there is at least a few long flights that still beat the 1:1 ratio, even if they don't run all year.

Icelandair flies a seasonal Reykjavik to Anchorage flight that takes 7h 10m, with a time difference of 8 hours, giving a ratio of 1.12:1

Condor flies a similar route, Frankfurt to Anchorage which takes 9 hours 50 mins for a 10 hour time difference, just squeezing in above even at 1.02:1

Many cargo flights also fly into and out of Anchorage and likely give even higher ratios, especially to places like Moscow (11 hours time difference, 4,385 miles direct flight distance)

Historically the record likely went to the Concorde, which flew between New York and Paris with a block time of about 4 hours, giving a ratio of 1.5:1 most of the year (6 hours time difference, daylight savings time dependent)