Let's consider a few issues.
There would be no atmospheric escape into outer space. This isn't much to begin with, but with the atmosphere being completely trapped, you have more to work with.
Next, The Earth's atmosphere is about 300 miles thick on the outside of a sphere. A sphere's volume is 4/3 pi*r3 The earth's radius is 6.37x106 meters for a volume of 1.08x1021 cubic meters. Add 300 miles (482,803 meters) and you get 1.35x1021 cubic meters for a atmospheric volume of 270x1018. If we process all that math to get the inner atmospheric thickness we get 359 miles.
Now, I've ignored the fact that the atmosphere thins as altitude increases, but this actually works in our favor as the thickness would be more then 359 miles due to the compression of density near the surface. Adding the effect of no atmospheric escape, and using an itch on my right elbow as my guide, I'm going to boldy declare that the thickness of your atmosphere is approximately 400 miles.
Finally, I'm going to note that, simplistically, the difference between having a sun that illuminates the entire world during the day and a sun that only illuminates half the world at any one time is that while on Earth you get winds, on your world you'll get a daily increase of barometric pressure and a nightly decrease. This will also have the effect of increasing the atmosphere thickness, which will reach its peak in the mid-afternoon. (I'm ignoring everything like seasonal conditions such as your sun switching on for longer periods during the "summer" and shorter periods during the "winter." Just one season for my answer, thanks!). How much the barometric pressure will increase (and therefore the atmospheric thickness) depends on how much the atmosphere heats and how much ocean you have to evaporate water into the air. But, I'm being outrageous, so let's claim that gets us to 450 miles of thickness at 3:00pm.
OK, you're trying to get to the diametric opposite of where you began. As Pojo-Guy points out, bouyancy will only get you so much altitude, but rather than worry about what kind of gas you're using and the weight of your dirigible, I'm going to estimate a best-case scenario.
As the crow flies, if a crow didn't need to breathe and its feathers were entirely heat resistant, we're talking about an 8,000 mile trip. But you can't reach an altitude of 4,000 miles.
What you can do is float straight up 450 miles in the afternoon, then proceed in a sinusoidal path (400 miles to 450 miles altitude depending on what time of day it is) until you're over your landing point, then you pull the rope and drop like a rock.
Thus, your fastest time would be the time needed to rise 425 miles (on average), (Tascend)) + the time needed to fall 425 miles (on average), (Tdescend) + the time to traverse the surface of a sphere 425 miles (on average) less than the surface the cows are standing on (and I'm assuming it's flat rather than worrying about whether or not we're starting/ending in a valley or on a mountain) (Tsphere), which is pi*r miles or 11,231 miles.
- You need to rise 425 miles.
- You need to fall 425 miles.
- You need to cross 11,231 miles.
If we assume the cruising speed of the Hindenburg, 76mph, and half that to ascend/descend, then it will take 22.37 hours to rise and fall, and 147.8 hours in transit for a hair over 7 days to make the entire trip.
Unless there's weather... Remember that nightly decrease in barometric pressure? Yup, rain. Lots of rain. Maybe even hail... the weather on your world will be interesting... and you'd hate one of those Far Side moments when your brother shuffles up next to you and touches your skin with a small hydrogen leak nearby....
And, lest we forget, you can't actually get that high. So the trip is probably a whole lot closer to 2 weeks.
You'll notice I'm using a dirigible, not a balloon. I'm not convinced you'll have significant winds in your hollow world. There's nothing pushing the air independent of the spin of the world other than the comparatively gentle push of evaporating and condensing water vapor. Methinks a balloon would be a curiosity as it can only ascend and descend, but not travel.