Is this sentence correct
Riding a horse is my hobby. I am very excited every time I ride.

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

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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

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

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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.