openvpn client connects to server but name resolution does not work

Solution 1:

Ubuntu does magic stuff with DNS, but fortunately OpenVPN provides a way to deal with that. If you've installed openvpn from the Ubuntu repositories, you'll have a script in /etc/openvpn/update-resolv-conf to tell Ubuntu what DNS it should use for your vpn connection. To use it, add the following lines to your openvpn configuration file:

script-security 2
up /etc/openvpn/update-resolv-conf
down /etc/openvpn/update-resolv-conf

That will run the script when your VPN connection comes up and goes down, and will update the DNS settings accordingly.

Edit (20-06-2018): As @Thenightmanager mentions in the comments, modern Ubuntu (at least 18.04 and newer) introduced some extra systemd DNS magic, and the update-resolv-conf script might no longer suffice. Check out update-systemd-resolved for a possible working alternative.

Solution 2:

It is worth noting that some form of NAT must take place for DNS (or any internet data) to work over a VPN. Typically: iptables -t nat -A POSTROUTING -s 10.8.0.0/24 -o eth0 -j MASQUERADE

However, You state:

iptables not installed

If this is so then your Server LAN router must perform NAT and have a static route for your VPN pointed at your VPN server IP.