What can oral baking soda do?
If your question is (which I don't think it is):
Does baking soda cure cancer?
The answer is that there is no support for that statement.
If your question is:
Can oral bicarbonate alter the pH of tumors in a mouse model?
The answer is yes.
This effect has been observed in this mouse model and replicated many times
This is an effect that has a been used for a while, and was first demonstrated in this 1999 article in the British Journal of Cancer. The addition of bicarbonate increases the buffering capacity in the live animal model, preventing the acidification of the tumor. It does not increase the pH of other tissues.
How does this work?
The idea here is not that adding a base increases blood or tissue pH directly, but it increases the amount of a physiologic buffer in a situation where that buffer has been depleted or is insufficient, allowing a better response to a pathologic excess of acid. Your question suggests you believe that an oral acid or base load will not change the pH of body tissues. This is true in normal physiology because there is a robust system of buffers. The primary extracellular buffer actually is bicarbonate. However, in diseased states, the buffer may become depleted or be insufficient. Oral (as well as intravenous) bicarbonate therapy is used in humans in several disease states, including certain forms of metabolic acidosis (Cecil Medicine, Ch 63, 120) and kidney disease (Ch 124, 128, 132). What is happening in these cases, and in the mouse model, is not a change in pH from the normal set point, but an increase in buffering capacity that allows the pH to return to the normal set point.
I would caution, though, that just because oral treatment is used in certain specific forms of acid base pathology, however, does not mean that there is support for the various forms of quackery that use arguments about flushing acids from your body, or using alkaline water as a general health tonic. It does address the question about whether it is very strange that orally administered buffer would impact a tissue with excess acid. It's not (strange).
The effect of changing the pH in the tumor microenvironment
The effect in the study behind the article you referenced, as well as others, is an increased sensitivity to cancer chemotherapy in otherwise resistant tumors. This is not horribly surprising, and doesn't mean by any stretch that baking soda cures cancer or that, e.g., an acidic diet causes cancer. Other studies, including this one (also in this very particular mouse model), show decreased invasion and metastases, also presumably because of the increased buffering capacity (and decreased acidifcation of the tumor and tumor microenvironments), so there may be other possible effects.
There has been a good deal of interest since 1999 in human studies adding bicarbonate to other treatments, but none have been published that I'm aware of, though I haven't checked the clinical trial registry.
On checking the clinical trial registry, on first glance it looks like some safety and tolerability studies have been completed, but it would take a fair amount to do a full analysis of what has been done and what hasn't. Maybe someone else wants to do it and include it in an answer? I'm fairly certain I would have seen and remembered an article showing that bicarbonate cures cancer in humans, though, so I don't think that's on the list :)