Is it possible to quest to 85 without a flying mount?

Expert Riding (225) skill is available at level 60 and allows flight in Outland. It costs 250g for the training, and 50g for a 225 level flying mounts.

Also available at level 60 is Flight Masters License, which costs another 250g. This allows you to fly in Kalimdor and Eastern Kingdoms.

At level 68 you can buy Cold Weather Flying for 500g that allows you to fly in Northrend.

Total: 1050g over the course of 25 levels. All of these costs are reduced by faction reputation with the city you buy the training in, and it's pretty easy to hit exalted with Stormwind or Orgimmar by level 60. This should be affordable just by vendoring the greens you get from quests and selling other materials on the auction house (meat, cloth, etc).

Now, to directly answer your question...

Outland is designed to be entirely land accessible, as originally you couldn't get flight until level 70.

Northrend is also mostly accessible on foot as it originally required 77 for Cold Weather Flying, but you will not be able to complete Sholozar Basin, Icecrown or Stormpeaks without a flying mount as many quest hubs are not reachable on foot. You are expected to be able to afford Cold Weather Flying by the time you finish Zul Drek.

Mount Hyjal can be completed entirely on a land mount, but you will spend a LOT of time running up and down the mountain.

Vashj'ir does not require a flying mount, as the entire zone is underwater. Shortly into the zone you are given a free underwater mount that is equally as fast as epic flight and is usable anywhere in the zone.

Deepholm will require a flying mount. The zone has many plateaus of varying heights that cannot be reached on foot.

Uldum and Twilight Highlands may be completable on foot, but you will spend a lot of time running around in hostile areas that you would otherwise be able to fly over and climbing up cliff faces.

In Conclusion...

You can get by with a land mount questing until about level 77, but after that your only option is running dungeons. This last patch removed the requirement to discover a dungeon before you can queue for it, so that may be a viable option.

However, unless you're really bad at managing your money, there is no reason you shouldn't be able to afford flying as you go. You don't HAVE to level your enchanting at the same time as your character.

There will be certain quests and quest chains you will not be able to complete without flying. But, you can level all the way to 85 without a flying mount, even if all you do is quest. You just have to be patient and willing to skip a few quest chains.