In order to install the bup backup software on Alpine Linux, you currently have to compile it yourself.
First, install the prerequisites:
apk add bash make g++ python3-dev git automake autoconf par2cmdline py3-pip && pip3 install wheel && pip3 install pyxattr
Now we can clone bup:
git clone -b 0.33 --depth 1 https://github.com/bup/bup
and build:
cd bup && ./configure && make -j4 install PREFIX=/usr
After this, bup should be installed in /usr/bin/bup. The bup clone directory we created in the first step is not needed any more.