The combination of an Apache-licensed program and AGPLv3-licensed library is covered by both licenses, meaning that both need to be followed. This does not change the license of the program itself - the library could be replaced. Somebody could take away the users' freedoms, but they would need to replace the library.
The Apache License 2.0 is compatible with the GPLv3 and AGPLv3 but not the GPLv2: https://www.gnu.org/licenses/license-list.html#apache2. Using an AGPLv3-licensed library in an Apache 2.0-licensed program is allowed, but you must follow the AGPLv3 when conveying it, which is incompatible with the Apple app store ToS.