Is it acceptable that I plot a time-series figure with years increasing from right to left?

I think readers will be strongly expecting that time increases from left to right in a graph. It's probably not a "rule" that you'll find written down anywhere, but it's certainly the overwhelmingly common practice. Having time go from right to left will very likely confuse your readers, and I don't think it should be done unless there is a very strong reason for it.

The "rule" is that you should create figures that make it easy for readers to understand what you are showing. That's because we use figures to convey information. So, if your choice of axis is confusing readers, then you've violated the rule. Looking at your figure, I find it confusing, and several of the others here appear to have had the same reaction.

So yes, in this sense, there is a "rule" that years should increase left-to-right unless you have a very specific reason to it the other way around, and that reason is to make something easier to understand -- for example, if you were talking about what someone would experience who is traveling backward in time.

As others have already pointed out: The convention is that the numbers should increase from left to right. And even though it's not really a rule, but only a convention, it is so common that any deviation might be hard to justifiy.

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However, it might be the case that you're just using inappropriate labels. You mentioned that your field is environmental science, but did not exactly say what the graph shows. For example, if your graph shows information about something like "How much of a certain substance that was emitted in year X can still be detected today", then the order might make sense. But then, the labels should be "The number of years that have passed", turning the absolute years into a duration, then being properly ordered:

^
|
|
+-------|----------------|------------...-----------------|----------->
  after 1 year    after 10 years      ...          after ~100 years
  (from 2017)      (from 2007)                      (from 1917)

(That's only a wild guess, based on wondering why you chose the "wrong" direction in the first place - but you might think about it...)