娇娇 早起 叫 舅舅 快 背 油篓 买 篓 油

The translation should be: Jiaojiao gets up early, asking her uncle to carry a basket that contains oil, and to buy a basket of oil immediately. It means Jiaojiao asks her uncle to do two things after she gets up: 1. carry a basket that contains oil 2. buy a basket of oil. Hope it helps!

" kuai bei you lou " is “ 快背油篓 ”，and " you lou " is on uncle's back.

I'm sure “篓油 ” 的 “ 篓 ” here is a measure word. Measure words are very complicated in Chinese, they're probably the most difficult part to learn for the foreigners. In Chinese, there're many measure words, some of them are often used, such as 个、条、双、座、台、本、件、辆、头、位、部、把、只、支、根 etc. We also have many measure words used not very often, I checked on line then I know that some of them borrow from nouns, they're called 借用量词，such as 一缸水（缸）、一碟花生米（碟）、一箱子书（箱子），so 一篓油（篓）is 借用量词 too.

Yes, in oral Mandarin, " 一 ” is often omitted，such as 给我来（一）杯茶、我想买（一）双鞋、他买了（一）辆车、抽（一）支烟、喝（一）口酒、他是（一）个学生、她戴了（一）顶帽子。So “ 买篓油 ” here is “买（一）篓油 ”， “ 一 ” is omitted.