I have not looked at the Arduino 101. Likely won’t for some time.
My first thoughts are
1) Not sure if the SdFat Lib work on it alone.
2) that its pin functions are different.
a) Interupts may be on different pins.
i) could try to use it in Soft Polling mode without interrupts.
b) The SPI are different pins.
So you may want to start with looking at these concepts first.
The SPI is the most important. The SdCardĀ and VS1053 both use it. So one needs to make sure it is wired correctly. Then need to make sure the SdCard works with the SdFat standard QuickStart.ino and VS1053 CS disabled.
If that works then the VS1053 should start to work.