Is there a way to secure the server again without the need of re-installing the OS?
If the device really was compromised the answer is: no
And of cause there is no way you can be a 100% sure that nobody accessed your system in this 24 hours but being sure is just the highest possible kind of probability and we can talk about probability.
When we asume that your potential opponent was quit an intellegent person you won't be able to prove that he did access your system even when he did. He might have altered every logfile, the bash history, the filesystem attributes and dates that might point out that something happened,... but these guys are pretty rare and often people forget to ask one of the most important questions: Am I a intersting Victim? If you have to answer this with no you will unlikely be a victim of this type of attacker.
Most attacks that happen on a technical level aren't carried out by highly skilled hackers. They are automaticly carried out by botnets or by scriptkiddies that scan millions of hosts every day and are lucky to get away with a handfull of compromised hosts. These opponents aren't that hard to detect and they also don't try hard to hide their activities because they want quantity not quality.
You might ask your Service Provider if any ssh connection has been established in the past 24 hours. If it's a proffesional provider they might be able to tell you. You can also monitor the processes executed and the traffic generated when you don't use the device the actively. But always keep the effort on an appropriate level.