You can arrive to the international airport called Carrasco, located in Montevideo, but it´s also usual to arrive from Buenos Aires by air or by ferry. There are several companies that perform the service and, some of them, offer one-day packages to enjoy the neighboring country.
Arriving to Montevideo by boat
Buquebus is the best known company and part of Puerto Madero, near Plaza San Martín in Buenos Aires. The company also offers packages for a day to Colonia and Montevideo, including tickets, excursions and lunch. There are two options: Boats direct to Colonia and Montevideo; or boat to Colonia and bus to Montevideo. If it is to spend the day, the latter is not convenient because it takes about 3 hours.
An alternative to Buquebus is Seacat, which is neither more nor less than the Buquebus sub-brand. Many times the boat they use is the same and the prices are cheaper.
The third option is Colonia Express, a company that starts its service from La Boca. It is the cheapest company among the three, but it also does not have a direct boat to Montevideo, since you have to go by bus from Colonia. It offers complete packages and, also, it can be paid from the web or at its offices in the city.
Arriving to Montevideo by air
This is the most obvious option, and the least exciting, too. Flights from Buenos Aires to Montevideo take 50-minute and operate mostly from Aeroparque Airport (AEP), rather than Ezeiza International. The route is dominated by Aerolineas Argentinas.