What follows is based on the notation example given, not a transcription of the audio.
The way your example is beamed makes it very hard to read the rhythm, because it obscures the beat, and because up/down strumming coincides with beats, it's especially hard to place the strumming motions.
Re-beam it first so you can see beats in 4/4 time. Strumming should be down on down beats and up between beats. But, because there are sixteenth rhythms, you should divide the up/down strokes again. In other words, put down strokes on the eighth note level, then use up strokes between eighth note divisions of the beat. Using violin bow direction marks for up/down...

...the rhythm then is...

To get the feel of the timing trying thinking of "silent" down strokes where you bring your strumming hand down without actually strumming the strings - swing your hand/arm just a bit above the strings. If those "silent" strums are put in parenthesis, it looks like this...

You could play those "silent" down strokes with an actual strum in the beginning, and then when you get the feel for the timing of the upstrokes make them truly silent. The end goal is to have your hand/arm moving evenly up/down, sort of like a pendulum and either strumming the strings or silent depending on the specific strum. Also, try counting the beat out loud as soon as you can coordinate it with the strum. That will really help tighten up your timing and rhythm.