There's this contest that I'm entering and I have to write something with only 1,000- words in it. Is it better to shorten phrases or to cut out details to meet the quota?
Although it is unclear from your question I'm going to assume you've surpassed 1000 words and are looking to reduce your word count.
Unlike other forms of writing, they key good fiction writing is 'voice'. If you read a good story you can 'hear' the narrator. In contrast if you read newspaper article or cookery book there's no voice - just clear and succinct information. If you remove unnecessary words and phrases from narrative you are messing the writer's voice, rhythm and natural flow.
The most common error of new writers is provide unnecessary information (particularly at the start).
I woke up in bed. I looked at the clock. It was 8:30. I was late.
-"I woke up." Everybody does that. "in bed" where else would you be? . . . this tell us nothing we need to know.
Compare: I pulled into the office parking lot.
"Johnson, you're late," said my boss.
What information is stated or implied in the second opening. (1) The character has a car. (2) The character has a job. (3) The character works in an office. (4) The character is late for his job. (5) The character's name is Johnson. (6) The character is a subordinate.
It's a more efficient way of writing.
Your sentences must have more than cosmetic appeal. What information are you imparting? Is the information relevant? Will it become relevant? Does the reader need the information now? What is stated as what is implied.
It's hard to say in advance, as both could be helpful. The best thing in my opinion would be to write first without limiting yourself either way, with the goal of reaching about 1300 words. Then, once you have this first draft, think about the best ways to shorten it. You will probably end up with some combination of the two approaches you described.
The answer is YES.
Focus on the essential: what happens. When in such a tight constraint, you really need to apply the rules of Chekhov's gun. In short, it means "if you put a gun in act 1, someone gets short in act 2" Theater speaks for if something is not directly linked to the action happening you don't need to mention it.
Personally, what I do is I write the story as a first draft to the point where I can say that was what I wanted to write. Then on that draft I will edit, either with an eye for cutting or adding words to get as close to the goal as possible. That way I work from something instead of worrying about what I need to do.
A few people love to peruse 400-page books, while others battle to peruse the initial 4 sections. Many individuals love to watch 2-hour motion pictures, while others would rather read a novelization of a similar story. A few people couldn't sit through a full-length motion picture, however could invest hours observing short recordings on the Web. To an ever increasing extent, as innovation has shockingly brought about many individuals having shorter abilities to focus, essayists are finding better approaches to impart stories in a way that interests to individuals who are accustomed to perusing short pieces of data on the little screen of their advanced mobile phones.