Python’s standard library, even in the latest version, does not include anything that can decode X.509 certificates. However, the add-on cryptography package does support this. Quoting an example from the documentation:
>>> from cryptography import x509
>>> from cryptography.hazmat.backends import default_backend
>>> cert = x509.load_pem_x509_certificate(pem_data, default_backend())
>>> cert.serial_number
2
Another add-on package that might be an option is pyopenssl. This is a thin wrapper around the OpenSSL C API, which means it will be possible to do what you want, but expect to spend a couple days tearing your hair out at the documentation.
If you can’t install Python add-on packages, but you do have the openssl command-line utility,
import subprocess
cert_txt = subprocess.check_output(["openssl", "x509", "-text", "-noout",
"-in", certificate])
should produce roughly the same stuff you got from your web utility in cert_txt.
Incidentally, the reason doing a straight-up base64 decode gives you binary gobbledygook is that there are two layers of encoding here. X.509 certificates are ASN.1 data structures, serialized to X.690 DER format and then, since DER is a binary format, base64-armored for ease of file transfer. (A lot of the standards in this area were written way back in the nineties when you couldn’t reliably ship anything but seven-bit ASCII around.)