Can I install series 24 in Precise 12.04

There is not a packaged prepared anywhere yet, but it should compile fine on 12.04.

Try this:
dget https://launchpad.net/ubuntu/+archive/primary/+files/deja-dup_24.0-0ubuntu2.dsc
dpkg-source -x deja-dup_24.0-0ubuntu2.dsc
cd deja-dup-24.0
sudo apt-get build-dep deja-dup
sudo apt-get install valac-0.16 devscripts
debuild
sudo dpkg -i ../deja-dup_24*.deb

That should likely work.

I just performed the steps you list above. It fails in debuild, I have pasted the last part of the build process below. I can attach the entire build log if interested.

make[4]: Leaving directory `/tmp/dejadup/deja-dup-24.0/tests/unit'
/unit/parse-dir: OK
make[3]: Leaving directory `/tmp/dejadup/deja-dup-24.0/tests/unit'
Making check in runner
make[3]: Entering directory `/tmp/dejadup/deja-dup-24.0/tests/runner'
make runner
make[4]: Entering directory `/tmp/dejadup/deja-dup-24.0/tests/runner'
  VALAC runner_vala.stamp
  CC runner.o
  CCLD runner
make[4]: Leaving directory `/tmp/dejadup/deja-dup-24.0/tests/runner'
/scripts/bad-hostname:
** (/tmp/dejadup/deja-dup-24.0/tests/runner/.libs/lt-runner:14194): WARNING **: runner.vala:215: Error string didn't match; expected (null), got Failed with an unknown error.
/bin/bash: line 1: 14194 Trace/breakpoint trap (core dumped) dbus-launch ./runner $test
make[3]: *** [check] Error 133
make[3]: Leaving directory `/tmp/dejadup/deja-dup-24.0/tests/runner'
make[2]: *** [check-recursive] Error 1
make[2]: Leaving directory `/tmp/dejadup/deja-dup-24.0/tests'
make[1]: *** [check-recursive] Error 1
make[1]: Leaving directory `/tmp/dejadup/deja-dup-24.0'
dh_auto_test: make -j1 check returned exit code 2
make: *** [build] Error 29
dpkg-buildpackage: error: debian/rules build gave error exit status 2
debuild: fatal error at line 1350:
dpkg-buildpackage -rfakeroot -D -us -uc failed

I get this error now:

 /tmp/dd dget https://launchpad.net/ubuntu/+archive/primary/+files/deja-dup_24.0-0ubuntu2.dsc
dget: retrieving https://launchpad.net/ubuntu/+archive/primary/+files/deja-dup_24.0-0ubuntu2.dsc
  % Total % Received % Xferd Average Speed Time Time Time Current
                                 Dload Upload Total Spent Left Speed
  0 0 0 0 0 0 0 0 --:--:-- 0:00:40 --:--:-- 0
100 1560 100 1560 0 0 19 0 0:01:22 0:01:21 0:00:01 2464
dget: retrieving https://launchpad.net/ubuntu/+archive/primary/+files/deja-dup_24.0.orig.tar.xz
  % Total % Received % Xferd Average Speed Time Time Time Current
                                 Dload Upload Total Spent Left Speed
  0 0 0 0 0 0 0 0 --:--:-- 0:00:40 --:--:-- 0
100 844k 100 844k 0 0 10563 0 0:01:21 0:01:21 --:--:-- 644k
dget: retrieving https://launchpad.net/ubuntu/+archive/primary/+files/deja-dup_24.0-0ubuntu2.debian.tar.gz
  % Total % Received % Xferd Average Speed Time Time Time Current
                                 Dload Upload Total Spent Left Speed
  0 0 0 0 0 0 0 0 --:--:-- 0:00:41 --:--:-- 0
100 7992 100 7992 0 0 97 0 0:01:22 0:01:21 0:00:01 30857
deja-dup_24.0-0ubuntu2.dsc:
dscverify: deja-dup_24.0-0ubuntu2.dsc failed signature check:
gpg: Signature made 2012-12-05T06:04:42 CET using DSA key ID 136B762D
gpg: Can't check signature: public key not found
Validation FAILED!!

Thanks Michael Terry, that solved my question.

I still get signing errors, but I just ignore them.