According to the image, it looks like you might have a recording that came with the book, so I would certainly suggest listening for how it is meant to be phrased.
Technique-wise, since this is a ukulele, you'll be finger picking everything. Commonly you'll see grace notes appear on the same string, which means you would use a hammer-on or pull-off by playing the grace note and immediately changing to the primary note--this also has the effect of not sustaining the grace note.
For notes on different strings like in this example, you will pluck both strings with different fingers, but you will pluck the grace note just before the primary note, and not sustain the grace note. (If the two strings are adjacent you could ostensibly pluck them with the same finger as long as you don't do it so quickly it sounds like a chord.)
In either case, the primary note should be rhythmically in time with the rest of the music, with the grace notes layered temporally on top of the existing rhythm. That's also one way to practice -- omit the grace notes until you can play the rest of the music correctly and in rhythm, and later add the grace notes in on top of the music you've already learned.