Hey! My husband and I did the Sundarbans trip from Kolkata last December, staying for 3 nights. We didn't do it as hardcore budget as you're planning, but we used public transport for the initial leg and saw a lot of backpackers doing the full public transport route.
From Kolkata, you can take a local train to Canning. It's super cheap, maybe INR 30-40 per person, and takes about 1.5-2 hours. From Canning, you'll find shared autos or vans to Godkhali. They fill up pretty quickly and cost around INR 50-70. This part is generally reliable; trains run frequently, and autos are always available at the station. Delays aren't super common for these main routes.
Once you're in Godkhali, things get a bit more local. You'll need to negotiate for a boat. Many backpackers form groups right there to split the cost of a private boat, or you can try to find a shared local ferry if you're not in a rush and your destination is a major village. The reliability here depends on the tides and the number of people. A full day private boat can be INR 2000-4000 depending on size and negotiation, so split between 4-6 people, it's manageable. A realistic daily budget for purely local transport (ferries, shared autos within the islands) would be INR 200-500 if you're good at finding shared options and aren't going too far. Just keep in mind that getting to truly "remote" areas by public transport can mean waiting for specific infrequent ferries.