I personally don't advise that you touch your version of php because of this question: "How to upgrade the server?" If you don't know the how, it is likely you don't know the why either and at that point... If it is not broken you may not want to fix it. That handles the last bit of your question up front.
Is it a good thing to upgrade? As long as it doesn't break anything.
Will it break anything? maybe. We don't know what kind of code exists on the server(phpmyadmin is probably safe but I suspect you have more). In practice you are most likely safe, but if you don't know what's going on you should probably leave it alone.
How to upgrade the server? It's been awhile since I played with apt but...
apt-get upgrade php There is a chance your repos won't have the latest versions. If that is the case and you NEED to upgrade... there are tutorials online... but you might consider bringing on a consultant.