For this to work on a first playthru you need the Master Key.
Run through undead burg, down havels tower and through Darkbasin and up to Andre and through to activate the elevators. Doing this allows you to skip the burg and the bridge.
When you go to Blighttown later you need to go through the backdoor (preferable anyway).
When you got the lord vessel, cleared demon ruins/izalith and killed the sunlight maggot (Solaire is still stuck in his starting position) you can go to him and activate him, he will now progress through his normal interaction points but you need to go to the bonfire in Anor Londo (the binfire in the wing with the silver knights).
After this he should be in the shortcut area in Izalith, sobbing about his sun without you having to open the shortcut.
Run through undead burg, down havels tower and through Darkbasin and up to Andre and through to activate the elevators. Doing this allows you to skip the burg and the bridge.
When you go to Blighttown later you need to go through the backdoor (preferable anyway).
When you got the lord vessel, cleared demon ruins/izalith and killed the sunlight maggot (Solaire is still stuck in his starting position) you can go to him and activate him, he will now progress through his normal interaction points but you need to go to the bonfire in Anor Londo (the binfire in the wing with the silver knights).
After this he should be in the shortcut area in Izalith, sobbing about his sun without you having to open the shortcut.