I don't think it's even possible to enter Lake Hylia without Epona unless you know the Serenade of Water in which case you probably would already have Blue Fire.
If you use any warping songs, like the Serenade of Water, the timed portions will time out. So that option's a no-go.
I think a run without Epona is possible, though you'd have to be extremely quick and know some good shortcuts to get to places on time. If you can get across the gates in front of Lake Hylia, and have the Longshot(not sure yet if the Hookshot works yet) to get across the Gerudo Valley's bridge, you should be able to do it. Just gotta do it very quickly.
You don't need any of that. Enter the crater from Darunia's chambers, and turn right toward the boulder that needs to break. Just roll jump directly to the left of the boulder and you'll jump just far enough to grab the ledge and climb up. Then climb the ladder and exit at the top. If you know where to jump from, it's actually faster than breaking the boulder.I expect that this shortcut might be faster, but you need either the longshot and the scarecrow's song or the hammer to do it: Zora's Domain -> Lost Woods -> Goron Village -> Death Crater -> Biggoron. In the crater, you either need to break a hammer-rock, or do a precise jump from the highest point down to the last scarecrow platform, and call the scarecrow to get off that platform.