Parker Solar probe launched in 2018 will take 7 years and multiple gravity assits to slow down enough to get close orbit to the sun.
It is also why BepiColombo will take 6-7 years to reach mercury orbit with similar steep delta-v costs.
Here is a delta V map [1] for the solar system. It would be easier to launch our trash to escape the solar system rather than land it in the sun.
Either way we generate nuclear waste in the millions of pounds per year , launch costs with everything spaceX is doing is nowhere cheap enough to even get the waste in significant quantities to even LEO.
[1]https://upload.wikimedia.org/wikipedia/commons/9/93/Solar_sy...
I don't have the exact numbers but roughly 9.08 μN/m2 is the radiation pressure @ 1 AU and depending on sail configuration (Square/Lattice etc) we can expect a λ ~ 0.25 , with a 800m2 and 5g/m2 density sail we can get effective acceleration around 1 mm/sec2, so you can reach near the sun in few(<10) years ignoring efficiency losses due to quartering and any payload weight etc.
In a real system you could speed this up a bit by using powerful lasers to improve acceleration and orbital methods like a cycler, but meaningful payload size would make it slower too.
The basic metric is that escape velocity of solar system is 30km/s earth starts you at 18km/s : it is easier to add 12 than drop by 18. You can do the same things at 2/3 delta-v budget and push your payload out of the solar system than into the sun.
[1] All orbits decay and eventually(10^150+ years) even the Earth will fall into sun ( or equivalent mass white drawf) so yes it is always possible
