I've seen just 'there's always a first' without 'time' and 'for everything'. Is it a standard usage?
"There's a first time for everything" is a standard phrase.
"There's always a first time for everything" also makes sense. There are many instances of that text if you do a search.
Conversely, "there's always a first" is not very common. We can try to invent a situation where that's the right expression. For example:
"Who was the first European explorer to sail in the Pacific Ocean?"
"Well... somebody had to be first. There's always a first."
In this case, "the first" is "the first person", not "the first time".
"Can I just say..."
No, they're not synonymous.