Trying to figure out the hours and timing for Mount Roberts tramway in Juneau, Alaska

You purchase tickets in the terminal in Tramway Station, 490 South Franklin Street, where you also begin the adventure. It operates from May through September and, as @Max points out, it posts operating hours on its Facebook page and is well-described by Wikipedia:

Base Facilities: The base terminal of the tram is located on the cruise ship docks on South Franklin Street. The terminal building features a large main hall, ticket windows, loading platforms with waiting areas, offices, and maintenance facilities for the tram.

Summit Facilities: The top terminal of the tram is located on a tower and offers spectacular views of the City of Juneau and the Gastineau Channel, Douglas Island and the community of Douglas to the west, the Chilkat Mountain Range to the north, and Kupreanof Island to the South.

The terminal platform is joined to the ridge of Mount Roberts by the Skybridge which leads to the summit facilities. The main building includes the Timberline Bar & Grill, the Chilkat Theater showing an informative film of the local Alaska Natives, and Raven Eagle Gifts. Located at the main building is the Stephen Jackson totem pole, a modern rendition of traditional Native themes. The Juneau Raptor Center operates a bald eagle display at the summit, which provides a recovery center for injured eagles and educational programs for tram visitors. Visitors can also enter the Nature Center which offers guided hiking adventures and books and maps on local recreation.

Several hiking trails of varying degrees of difficulty (including wheel-chair accessible paths) have been laid out leading from the summit facilities. Many of these feature spectacular views from the Mount Roberts ridgeline, and some wind through the forest trees and meadows with wild flowers and animals. The forest paths feature trees with totemic carvings depicting Native legends. Interpretive markers describing many of the flowers, plants, trees, birds and animals are placed along the loop trail for self-guided walks.

The Juneau Raptor Center operates a bald eagle display at the summit, which provides a recovery center for injured eagles and educational programs for tram visitors. Visitors can also enter the Nature Center which offers guided hiking adventures and books and maps on local recreation.

Several hiking trails of varying degrees of difficulty (including wheel-chair accessible paths) have been laid out leading from the summit facilities. Many of these feature spectacular views from the Mount Roberts ridgeline, and some wind through the forest trees and meadows with wild flowers and animals. The forest paths feature trees with totemic carvings depicting Native legends. Interpretive markers describing many of the flowers, plants, trees, birds and animals are placed along the loop trail for self-guided walks.