Yes, Austin is the capital of Texas and it sits squarely on the Colorado River—but there's a catch. The Colorado River flowing through Austin is not the famous one that carves the Grand Canyon. That's a completely different river system located over a thousand miles to the northwest.
The Texas Colorado River is the longest river wholly within Texas, stretching over 860 miles from the Llano Estacado to Matagorda Bay on the Gulf of Mexico. Austin's location on this river was strategic for early settlement, providing water, power, and a transportation route. Today, the river is dammed to form the Highland Lakes (including Lady Bird Lake right downtown), which are central to the city's recreation, water supply, and identity.
The naming overlap is a quirk of history. Spanish explorers named the Texas river "Colorado" (meaning "red" or "reddish") for its silt-laden waters. The Grand Canyon river was named later by different explorers. While Austin's river is vital to the city's existence, the prompt's premise that a capital is "important for the state of Texas" *because* of its river location is slightly misleading. Austin's importance stems from being the seat of government, a tech hub, and a cultural center—the river is a geographic feature it was built upon, not the sole source of its significance.