How can I downgrade openssl via apt-get?

I do not believe that a downgrade will solve your problem. But you asked:

The package openssl=1.0.1-4ubuntu5.31 isn't longer in the Precise repositories, because the latest version is now 1.0.1-4ubuntu5.32. But the package is still available in the PPA of the “Ubuntu Security Team” team.

For version 1.0.1-4ubuntu5.31

wget https://launchpad.net/~ubuntu-security/+archive/ubuntu/ppa/+build/7531893/+files/openssl_1.0.1-4ubuntu5.31_amd64.deb
sudo dpkg -i openssl_1.0.1-4ubuntu5.31_amd64.deb

But I suspect that you have to downgrade other packages listed here.

The downgrade is temporary. After a

sudo apt-get upgrade

or for a better feeling a

sudo apt-get dist-upgrade

the latest version will we installed again.

According to https://launchpad.net/ubuntu/+source/openssl, there is no version 1.0.1-4ubuntu5.31 of the package openssl in any 12.04 (Precise Pangolin) repository any more.

The only available versions of openssl for 12.04 as of today (2015-12-30) are:

• 1.0.1-4ubuntu5.32 from the main repository's updates and security channel (2015-12-07)
• 1.0.1-4ubuntu3 from the main repository's release channel (2012-04-19)

The 1.0.1-4ubuntu5.31 version you want to downgrade to was replaced and is no longer available. You can only downgrade to 1.0.1-4ubuntu3, if you don't mind a 3 years old version (which probably contains a few serious bugs and security holes!) or find a PPA hosting a not that outdated version of openssl.