Adding a small straw to allow air into the top of the sphere is a good idea. This will allow air to flow in and water to flow out simultaneously. You may need to work to find a straw small enough. Another option to fulfill the same purpose would be a glass capillary tube.
If the sphere is completely full you will need to first drain some of the liquid. Then you need to get the straw through the water without it filling with water. So you'll need to cap the end of the straw to trap in air and push it through the water. Then uncap the end and it should allow air to go in while water goes out.
The problem you will run into is that the holes are so small that the surface tension of the water will greatly hinder the flow. Adding a small amount of a surfactant, such as a detergent, will significantly reduce the surface tension of the water, allowing it to flow more freely through the small opening. Combine that with the straw and it should work well.