It depends on how the server is accessible, and whether it's an actual target anyone is trying to actively hack(as opposed to various bots that that try to log in as root with password of "password" that exclusively hit ports 22 and 2222).
Firstly, a key concept here is attack surface. If you only have SSH and Apache with a static website running, and all other ports are blocked by firewall, then there just isn't much that can be exploited even if there are vulnerabilities that haven't been patched yet. If you have PHP, SQL and Wordpress running, then suddenly there are a lot more attacks that can be successfully executed if any of them are vulnerable. And as the attack surface grows - as you have more programs running and more services that can be accessed(directly or indirectly) - the probability of there being an exploit that can be used to target your server grows.
Secondly, you have to consider the threat model: Who is trying to hack into your server and why? Apart from the aforementioned bots(which, as noted, are generally foiled just by following reasonably good password practices and not using default usernames like root, admin, etc.), who would try to get into your server? Hacking people takes time, and most people who do it just for fun prefer doing it legally. If you have a personal website that gets a couple dozen visitors per day, there isn't much to be gained from hacking it(if, on the other hand, you're running the website for a large corporation, you need to seriously consider the threat of someone spending a lot of effort to gain access to your system).
Thirdly, updates are typically not as time critical as you seem to think. Yes, there are zero-day exploits in the wild and sometimes a vulnerability can be exploited very quickly, but even if a vulnerability is known it takes time to develop an exploit - and in many cases, the exploits will only be known by the developers and a few security researchers until a patch has been developed and is ready to be deployed. So even if you don't patch instantly, it's unlikely that anyone could exploit the particular vulnerability within a few days(unless it's one for which a zero-day exploit is known).
In conclusion, while it's almost certainly possible for your server to get hacked within a few days, as long as you follow reasonable security practices(and aren't being specifically targeted by malicious actors) it's very unlikely to happen just because it took you a day or two to apply the latest security patches.