Grout cracked & discoloured
"I rather want to know, whether I need to worry and act immediately or can lay off the fixing for a while."
In answering this specific question it's obviously not going to get any better without remediation. However, it isn't an emergency situation as long as you can keep that area dry. If it's just you, that might be workable but if you have a wife and three kids using it- good luck. Even small amounts of moisture seeping under that tile may cause mold, mildew and possibly other issues.
So if it's a landlord issue - why put it off?