The problem is less likely the security of the Ubuntu system as shipped by the distributor. It is very rare that there are critical issues which allow an remote attacker to hack the system. And while there are sometimes critical issues which allow privilege escalation by a local user they are rare too. But you can actually limit the impact further by only installing the software you really need - anything which isn't there cannot be exploited.
What you see as security updates are almost always less critical problems, like exposing address layout which might be used to bypass ASLR, security updates for software you don't even use but have installed, updates for issues which happen only in specific environments or with non-standard configuration, updates for issues which allow applications to crash (denial of service) but cannot be used to take over the system etc. See usn.ubuntu.com for the details. Anything which might help an attacker to advance is considered a security issue but in most cases the attacker needs to be local already or can only do uncritical harm.
But in no way you should feel secure just because you've applied all security updates coming with the system or feel totally insecure just because you've missed the once from the last days. It is more likely that a hack will be done be exploiting issues in your own web application you've deployed on the server or that you've used weak password to protect parts of your system. It is actually the most common way that system gets owned through thus user-generated weaknesses and not through weaknesses caused by the underlying OS.
Thus, while you should keep your system up-to-date you should not blindly rely on these updates keeping your system secure. Additionally you should make sure that your own applications and configurations are secure too. This aspect might even be more relevant in securing the system than keeping the system always up-to-date.