Look for named.conf (usually /etc/named.conf) on the existing server. It will tell you which zones the server is authoritative for, and which files contain information for each zone.
Ubuntu will have a bunch of configuration files in /etc/bind. The configuration has been split for ease of use, so you may want to have a read at these first rather than just copying the files over.
Zone file syntax hasn't changed much over the years, but there may be one or two options in named.conf that need revising. Your best bet is to stop your new BIND server (BIND9?), put the old server's config in place (remembering to back up the existing config first!), then try starting BIND and see what happens.
BIND logs to the system log, so you'll see information appearing in /var/log/syslog (or possibly /var/log/daemon.log, depending how syslogd is configured).
If you get stuck, there's an excellent BIND9 configuration reference here:
http://www.bind9.net/manual/bind/9.2.5/Bv9ARM.html
Have fun!
S.