My cat seems to be sleeping longer now that he's getting older.
My cat spends more time sleeping because he's getting along in his years.
この場合、次のような言い方ができますよ。
ーMy cat seems to be sleeping longer now that he's getting older.
「うちの猫は歳をとってきて寝てる時間が長くなってきているようだ」
to get older で「歳をとる」
ーMy cat spends more time sleeping because he's getting along in his years.
「うちの猫は歳をとってきてもっと寝るのに時間を費やす」＝「寝る時間が多くなった」
to get along in years で「歳をとる」
ご参考まで！
カナダ 
