Your sentence is grammatically correct. The issues others raise are points of style that make your writing more appealing to a reader.

My hobby is horseback riding. I get so excited every time I'm riding a horse. Please contact me if you'd like someone to take a look at your sentences and how we could improve them

Está correto, porém em vez de "montar um cavalo", que soa estranho, a palavra cavalgar ou equitação seria mais correto como substituição.

Horseback riding is a hobby of mine. It’s exciting every time I ride. The second sentence is a bit different from yours, however, it’s less stunted.