A 5 mm stone in the ureter can sometimes pass spontaneously, particularly if it is in the lower/distal ureter. However, your report showing mild hydronephrosis means that the stone is causing some degree of urinary obstruction, so it should not be ignored.
There is no reliable way to confirm at home that the stone has passed. You may see the stone in a urine strainer, but even if your pain improves, the stone may still be present. Follow-up imaging, usually ultrasound KUB as advised by your doctor, is needed to confirm that the stone has passed and that the hydronephrosis has resolved.
If you have no fever, infection, significant
kidney-function abnormality, uncontrolled pain or worsening obstruction, a short period of conservative treatment may be reasonable for a 5 mm stone. Tamsulosin 0.4 mg once daily is commonly used as medical expulsive therapy, particularly for stones in the distal ureter, but it should be taken after discussion with your treating doctor. It may cause dizziness or low blood pressure.
Alkasol/potassium citrate is not required for every ureteric stone. Its usefulness depends on the type of stone and urine findings, so it should not be started routinely without medical advice.
Homeopathic medicines such as Berberis vulgaris have not been proven to reliably dissolve or expel ureteric stones, and they should not replace evidence-based treatment.
Maintain normal hydration, but do not force excessive quantities of water in an attempt to flush the stone out.
Because there is already mild hydronephrosis, I would recommend review by a urologist, particularly if the stone does not pass over the expected observation period or if hydronephrosis persists.
Seek urgent medical attention immediately if you develop fever/chills, worsening or severe persistent pain, repeated vomiting, difficulty passing urine, or marked weakness. A urinary infection associated with an obstructed kidney can be an emergency.
If the stone fails to pass or obstruction persists, the urologist may consider ureteroscopy or shock-wave lithotripsy, depending on the stone's exact location, anatomy and other clinical factors.
If you upload the complete ultrasound/CT report, especially the exact location of the 5 mm stone, degree of hydronephrosis, and any urine/kidney-function reports, I can help you understand whether conservative management or earlier urological intervention would generally be more appropriate.