Direct bus from Johannesburg Airport to Mbabane

From Wikitravel's Swaziland article:

Most public transport bus services arrive in Mbabane or Manzini. Smaller bus lines, or minibuses generally provide service to Johannesburg, Durban or Cape Town in South Africa as well as Maputo in Mozambique.

Larger buses usually travel within the country and some stop at border crossings, where passengers must connect with an onward journey, unless a specific group booking is done to hire a big bus.

For scheduled road transport there is the Swaziland based siyeSwatini TransMagnific, which provides transport to and from Swaziland daily. Stops include the Johannesburg airport. The TransMagnific mini-buses are customised for added comfort and safety, unlike the public transport. However, they require that bookings and payments be done at least a day prior to travel so that your meal can be ordered and the selection for the movie can be determined for the +-4hour trip.

The South African Baz Bus, an independent line targeting backpackers, also makes regular stops via South Africa to various hostels and hotels in Swaziland. When traveling into and out of South Africa to and from Swaziland, this is the safest option. All mini-buses into South Africa go directly to Johannesburg bus stations, which can be dangerous.