To begin construction of a hotel at GEO we construct a new tether, to catch payloads from the LEO tether. The first tether will have ballast of about 60 tons, with some solar sail to rebuild momentum shared with a catch. The LEO tether will forward a payload consisting of 3 tons of useful cargo and a 1 ton solar sail into GTO. (The sail will unfold after toss, at an altitude of 1000 km, so avoid drag.)
The GTO orbit needs another 1.4 km/s at GEO radius to circularize the orbit. As a quick plausibility calculation; to give 4 tons a DV of 1.4 km/s will require 5.6 million Newtons. A 1 square kilometer solar sail has a force of about 8 Newtons, when the velocity vector is away from the sun. The GTO orbit will be chosen such that the apogee has this condition (overhead at sunset). An eccentric orbit spends most of it's period near apogee, we use 50% useful force for an approximation. So 1 1 km solar sail would need 1.4 million seconds, or about 16 days to provide that momentum. As the orbit is circularized, a smaller percentage is available, so we allow one month for circularization of the cargo orbit.
A payload in a not-yet circularized orbit (the next payload, for example) will have a shorter period, and will pass a circular-ized cargo with a low relative velocity. If the first cargo has a tether and a grapple, it can catch succeeding payloads and use all accumulated sails to provide momentum for later arrivals. This compound growth will compress the time to gather enough ballast to begin a rotating tether operation.
After the initial 60 tons of ballast are gathered, payloads will be winched in, which will increase the rotation rate (due to conservation of angular momentum). As the tip of the tether gets to 1.4 km/s, payloads can be accepted directly from GTO, so cargo will not need to provide its own solar sail.
The GEO tether will have two central masses, the hotels, about 20 km apart. A 100 km tether (80 km beyond the end of each hotel) will reach out to catch tourists, and winch them in to the hotel. These two winches will be used to continue the construction of the hotel.
As segments of the GEO hotel are available, they will be installed. Having mass at a radius, about 20 km, will store angular momentum, so this tether will not speed up very much with additional payloads. I.e. the increment of angular momentum will be a smaller fraction of the existing angular momentum of the hotels.
We need to manage angular momentum to keep the hotels from spinning faster with each payload. We expect to get one payload a day during construction. If the winch pulls in payload while the GEO tether is horizontal, the addition of angular momentum will be reduced. Another winch to let out some mass when the tether is vertical and pull it in when the tether is horizontal can reduce angular momentum. Since the payload can take two days to get into position, these two processes can add to each other.
The GEO Hotel has a design mass of 2000 tons. At 4 tons per day, it would take 500 days to put that much mass into GEO. If the LEO tether adds some solar cells, it could provide momentum to toss 8 tons a day, reducing the construction time to 250 days. This would require raising the LEO tether to about 8.5 megawatts of PV.
When the hotel has enough rooms for some guests, and staff to support them, we can start accepting paying tourists. For example, when we have built 1/7 of the hotel, we could keep the LEO tether busy if we only allowed guests to stay overnight. This could bring in cash flow to help pay for the rest of the hotel.