First things first: write your assessment objectives in really big letters at the top of the page. This will help you to remember to include them in your essay because when you're under pressure, it's easy for them to slip your mind. Next, read the unseen extract three times. This sounds like a lot given the time pressure, but it will definitely improve your essay because its only after reading something three times that you will really be able to comprehend it and pick out the nuances of the extract. When you're reading through the extract the first time, make a quick note next to each paragraph summing up the main idea expressed in that paragraph. Not only will this help you with comprehending the prose extract, but it will help you navigate the extract once you start to analyse it.
Once you've read through the extract once, read through it again, underlining or highlighting the parts of it that jump out at you as being interesting. After this, group these underlined parts into themes. These themes could be anything, but are most likely going to be things that you've studied to do with the time period that the extract is taken from such as politics, attitudes towards women, class etc. Go on to writing your essay, starting with your first theme. As you go through the essay, make sure that you're getting in the assessment objectives that you wrote down at the top of the page, especially close language analysis. This method of analysing should help you get through the unseen prose exam analysis quickly and successfully.