Algorithm for scheduling unit time tasks with arrival times and deadlines
The more general version of your problem is the scheduling problem where we have $n$ independent tasks, each with an arrival time, a resource requirement (time), and a deadline. We define the Earliest Deadline First algorithm such that at every timepoint, we choose to process the task with the nearest deadline. While time is continuous, it is trivial to see that we can discretize time to be all points at which a task finishes or a new task arrives. 

It is known that EDF is an optimal uniprocessor scheduling algorithm i.e. if there is a feasible schedule, EDF will find it and if EDF does not find a feasible schedule, then no feasible schedule exists. [1], [2]. However, this is not linear time if we aren't guaranteed an ordering of the deadlines of the arriving tasks. Let's use a priority queue to represent the waiting tasks. At every time point, which there are $O(n)$ of, we must insert and/or pull the next task from a priority queue sorted on deadlines. Each insertion is $O(\log n)$ time, thus resulting in $O(n\log n)$ time. 

If your input was sorted by arrival, we could easily find whether or not there is a valid schedule by following the algorithm above, iterating through the days, adding tasks for that day to a priority queue and selecting the task with the nearest deadline from the queue. We fail if we ever try to assign a task to a day on or past its deadline. 

As mentioned in the comments, we can sort the input by arrival time in linear time using counting sort or pigeonhole sort. The issue then is to somehow make the priority queue also linear time.