I think you can say for example: Die Liebe meines Lebens, this means the love of my life. But can you use 'meines Lebens' with a noun different from 'die Liebe', for example: 'Die Zeit'?
You're probably thinking: Can I translate "the time of my life" as "die Zeit meines Lebens"? The answer is no, for three related reasons.
- The construction "meines Lebens" to mean "the greatest/highest/best/... I've ever experienced" is semi-idiomatically linked to "Liebe". It also works with "Mann", "Frau" and some others, but not with "Zeit".
- "Zeit" doesn't mean "kind of experience" in the first place. Germans will understand "Eine gute Zeit haben", but it sounds like an awkward anglicism rather than normal German (like many other anglicisms, this one may eventually become more accepted).
- There already is another very similar idiom that interferes with this one: "Zeit meines Lebens" means simply "throughout my life" (with no superlative meaning). This is a rather particular construction with an absolute noun used in an adverbial function, and it is probably another barrier to "Die Zeit meines Lebens" being used to mean the best time.
The phrase »meines Lebens« is just the genitive case of »mein Leben« (my life). You use it whenever you talk about »my life« and when the grammatical construction of your sentence requires genitive case for this phrase.
Zu Beginn meines Lebens war ich ein Baby.
At the beginning of my life I have been a baby.
Die Ereignisse meines Lebens haben mich zu dem gemacht, was ich heute bin.
The happenings in my life made me to what I am today.
»Über die Dauer meines Lebens möchte ich nichts sagen«, antwortete die Diva im Interview.
»I don’t want to say anything about the duration of my life,« answered the diva in an interview.
There are some phrases, that contain »meines Lebens«, which have a special meaning:
Ich war zeit meines Lebens auf der Suche nach jemandem wie dich.
I was during my whole life looking for someone like you.
Here the word »zeit« is not a noun but a preposition meaning »during the whole« and can only be used together with the noun »Leben« which has to be used in genitive case. This word can not be used together with any other period of time (»Ich war zeit der ersten drei Unterrichtsstunden noch sehr müde.« doesn't work)
Die Jahre, die ich in England verbracht habe, waren die Zeit meines Lebens.
The years, that I spent in England, was the best time in my life.
Here »Zeit« is a noun, and depending on the context »Die Zeit meines Lebens« can mean »the best time in my life«. In the song »Zeit meines Lebens« my Martin Stosch this phrase is used exactly in this meaning. But within a different context it can also just mean »a period of time within my life«:
Die Jahre von 1984 bis 1990 waren die Zeit meines Lebens, in der ich Medizin studiert habe.
Martha war die Liebe meines Lebens.
Martha was the greatest love of my life.
»Die Liebe meines Lebens« is in almost every context the great love of someone's life. You can also replace »Liebe« with »Mann« or »Frau«, depending on the gender of the person who is loved:
Neil sagt über seinen Ehemann: »David ist der Mann meines Lebens«.
This means, that David is the great love in Neil's life.
To comment on your example: No, you can't use
Zeit meines Lebens as noun.
You can say:
"Zeit seines Lebens war er nie in Rom." which could be somewhat translated to
"During his lifetime he has never been to Rome." or another German example:
"Zu Lebzeiten war er nie in Rom." Maybe he's burried there now.
Zeit in this context would be a genitive preposition and therefore is written in lower case (when not in the beginning of a sentence).
More on this in this german article.