Student or Learner
Definition of jump, Macmillan Online Dictionary: Free American English Dictionary and Thesaurus
The Japanese, once again, may be embarrassing the US manufacturers by getting the jump on them.
Cambridge Dictionaries Online - Cambridge University Press
If I leave work early on Fridays I can get a jump on the traffic.
Are there any occasions where either "get the jump on" or "get a jump on" sounds more appropriate than the other?
As for the second sentence, I would leave it at "get a jump on". It seems logical to understand this advantage - headstart - jump on - as one of any number of advantages.
For it to make a difference, we would have to clearly see that there is a specific advantage that one is referring to when using "jump on" to mean "advantage". If there is clearly a specific advantage, then use "the", of course.
Articles can be very tricky for some ELLs. What makes it more challenging is that sometimes there is more than one correct answer, and to justify more than one correct answer takes some explaining, explaining which may only be more confusing in the end.
I have lots of article exercises that I've collected from the Internet over the years, as well as documents that outline explanations for English article use. I would recommend lots of practice doing exercises and reading texts. Read and underline the articles and the nouns they precede, and then attempt to explain why an article is used before each noun. Underline all nouns that don't have an article and explain why there is no article. This would be worth it for those that really want to use articles correctly and are having a difficult time with it. Of course, exposure to the language and observing is helpful, but observing more closely would make sense if leaving it to "acquisition" seems not to be working very well.