Telemann spent a large part of his compositional career writing for pupils in lieu of his teaching position. With this in mind, it is perhaps more important than usual to investigate a range of Telemann’s music, for its non-programmatic elements can be imitated by listening to common traits in the style.
In all honestly, I doubt this piece is ideal for a first work for a student starting piccolo trumpet. While the range can often be achieved relatively quickly, a great sound in the upper register and ability to play with finesse typically takes longer. I would recommend Phil Collins’ etude book for piccolo trumpet; spending time making a great sound in the middle register will create fundamental qualities that can be transferred to the upper tessitura.
Musically, Telemann’s trumpet writing should, I believe, imitate the lightness of articulation of a flute. Slurring every other sixteenth in running passages or slurring the first two sixteenths only can offer this effect. Be sure to KNOW the direction of each phrase and communicate with your collaborators on the harmonic and melodic direction of each line.