The YouTube video you have link shows someone who has become unfortunately famous in Italy for being a fraud (sorry, some of the sources are in Italian, but Google Translate should suffice for a basic translation).
Tullio Simoncini was a medical doctor who proposes a pseudoscientific theory by which tumours would actually be derived by a fungal infection (specifically from Candida albicans) and that should therefore cured with sodium bicarbonate. His theory is absolute pseudoscience, as there is no evidence for tumour being generated by Candida infection, and his treatment is nothing more than a very dangerous hoax to spill money from desperate people.
In 2003 Simoncini had his license to practice medicine revoked and was condamned for fraud and involuntary manslaughter after the death of three people due to his treatment.
In 2007 a breast cancer patient in the Netherlands died after the same treatment.
On 17 January 2012 Italy's Supreme Court (Corte di Cassazione) has finally confirmed his condemnation for menslaughter (sentence 1432/2012) (source, in Italian).
He has also been recently sued in Albany by a family of a brain cancer patient who died after Simoncini's bicarbonate treatment (source, in Italian).
Now, although Simoncini's theory is pseudoscience, this does not mean that sodium bicarbonate could not have an effect on cancer.
There has been some scientific research on the effect of bicarbonate on tumours. Due to their high metabolic rate tumours are, in fact, acidic (pH ~6.5 compared to ~7.2 in normal tissue), and bicarbonate can act by increasing the pH in the tumoral environment. This could be important because low pH seems to be an important factor in chemoresistance and could have some effect in promoting metastases.
This study shows an important effect of bicarbonate in preventing metastases in a mouse model of breast cancer:
Bicarbonate increases tumor pH and inhibits spontaneous metastases - Robey et al. - Cancer Res., 2009
However, the authors warn that:
Medically, the idea of treating cancer through p.o. administration of buffers is attractive but tethered to caveats. Reaction diffusion models show that the effect of bicarbonate on the pHe gradient will be graded with dose and that, at the current dose of 200 mmol/L, is not saturating. This dose translates to an intake of ~1.5 μmol/h/g of whole mouse. By comparison, the acid production rate of tumors can be ~100 μmol/h/g of tumor weight. Thus, these doses of NaHCO3 may be able to counteract the acid load of a 15-mg tumor, which translates to ~ 105 cells or a 1-mm3 micrometastasis. The effectiveness of this therapy will be reduced with larger tumors. It is somewhat surprising that this incomplete effect had such a dramatic effect on metastases. Another concern is that bicarbonate, with an effective pKa of 6.24, would seem to be poorly suited as an alkalinizing buffer; thus, it is possible that better pHe control and more dramatic antimetastatic effects will be observed with a higher pKa buffer. However, it remains possible that this effect may be specific for buffers in the bicarbonate/CO2 family through involvement of carbonic anhydrase activity, which is important to pH regulation in tumors.
This is also noted in this other study by the same authors
Certain determinations are required before considering systemic buffering of tumor acidity as feasible approach to cancer treatment. Effective doses, susceptible tumor types, ideal buffers, safety approaches, and other factors in the tumor microenvironment would need to be identified and validated. To assist our understanding of these phenomena, mathematical models were developed to predict the safety and efficacy of systemic buffering of tumor acidity in humans. Our findings suggest that chronic use of oral bicarbonate as a cancer intervention is limited. Safe doses for consumption limit the amount of buffering to counteract tumor cell production of extracellular protons. In a previous study, it was reported that the saturating dose of oral bicarbonate in mice (which roughly translates to about 0.18 g/kg/day in humans) was only sufficient to counteract the acid load of a 15 mg tumor consisting of about 100,000 cells or 1.0 mm3 . Moreover, chronic application of oral bicarbonate at doses higher than 0.5 g/kg/day is predicted to induce systemic alkalosis (Martin N, Robey I, Gaffney E, Gillies R, Gatenby R, Maini P: Predicting the Safety and Efficacy of Buffer Therapy to Raise Tumor pHe: An Integrative Modeling Study, submitted).
To summarize, sodium bicarbonate alone is not going to cure cancer, and at high doses it can even be dangerous.
There may be uses of bicarbonate or other pH modulators to enhance chemioterapy approaches, by increasing their efficacity thus allowing reducing the doses and the secondary effects.