Your base facts are off. It is not the diameter of the Earth that is important, it is the Earth's circumference. At the equator, the Earth's circumference is approximately 40,000 km. Thus, to achieve what you suggest, you would need to travel at about 1,700 km/hour or 1,100 mph. This is much faster than any current commercial aircraft. This is also Mach 1.4 (i.e. faster than sound).
You could, however, circumnavigate the Earth closer to either pole, making your journey much shorter. The timezones do reach all the way north and south, at least in theory. Close enough to the pole and you could walk!
You do run into another issue though, and that is that the timezones are not uniform in size. Some are much larger and their size depends on your exact latitude. So, while you might be able to keep the sun in a fixed position in the sky (relative to yourself), the local time may jump around a fair bit.