Can I update the Beaglebone Black without a micro SD adapter?
If your BBB is functioning, and you want to write an image to a microSD card, then yes, you can do it with the BBB.

You can place your image, as a file, on any USB storage device (formatted as MSDOS / FAT32 / VFAT / whatever you call it).  Then mount that device, and use DD to write the image to the SD card:

$ sudo mount /dev/sda1 /mnt
$ sudo dd if=/mnt/myimage.img of=/dev/mmcblk1 bs=32k


Of course, make sure you use the right device names for your USB storage and the SD card - you don't want to accidentally DD over the eMMC...