Shy is not something we can easily avoid becoming; genetics and life experiences both contribute to it, along with how we are raised. Added to the fact that many shy people don't realize there are ways to help it - shyness then gets thought of as a non-fixable personality trait. Even if a shy person does know there are ways to help, shyness by its very nature could keep them from asking for it.
This article at Psychology Today explains shyness well, including the contributing factors:
According to developmental psychologist Jerome Kagan, Ph.D., and
colleagues at Harvard University, up to a third of shy adults were
born with a temperament that inclined them to it. The team has been
able to identify shyness in young infants before environmental
conditions make an impact.
Further evidence of a biological contribution to shyness is a pattern
of inheritance suggesting direct genetic transmission from one
generation to the next. Parents and grandparents of inhibited infants
are more likely to report being shy as children than the relatives of
uninhibited children, Snidman found in one study.
Most shyness is acquired through life experiences.
How we're raised:
Some people are born with a temperamental tilt to shyness. But even
that inheritance doesn't doom one to a life of averting others' eyes.
A lot depends on parenting.
The incidence of shyness varies among countries. Israelis seem to be
the least shy inhabitants of the world. A major contributing factor:
cultural styles of assigning praise and blame to kids.
But shyness can sometimes be overcome, at least to some degree. The article also states:
Despite the biological hold of shyness, there are now specific and
well-documented ways to overcome its crippling effects.
As mentioned by Kev Price in the comments, CBT (Cognitive Behavioral Therapy) has been shown to help. I'm not sure if you would have the chance to incorporate a bit of that into a one-week visit, but it might be something to suggest to your friend. Here is a good overview of what CBT is and how it can help.
In reference to Catija's comment above: being unable to speak to people when you are shy is very normal. That characteristic is practically the layman's definition of shyness. We have phobias to describe the extreme cases: fear of leaving your home, fear of the public, etc.
To answer your question:
The list of starting activities sounds like a good beginning. The main things to remember are:
- Start small
- Always have their back
The second point can mean anything from not leaving them to fend for themselves at a crowded party, to cheerfully leaving that crowded party with them 5 minutes after you arrive because it's too overwhelming for them.
Also bear in mind that your friend might agree to go do a certain activity, but back out when the time comes. This is not motivated by anything in particular (except sometimes, panic) - most of the time it's because the shy person thought, "Sure, I can do that," but as the time gets closer they realize how awkward, overwhelming, or far out of their comfort zone it might be.
If this happens, make sure they know you are not blaming or judging them, and see if you can talk through what about the activity is causing the reticence or distress. Don't try to debate their points, just listen. If they are willing, you two can try to think of another activity to replace that one with.
As time goes on and your shy friend starts to trust in the fact that you are not going to judge them, put them down, talk them into doing anything, or tell them they need mental help, that trust is likely to give them more confidence to try new things. Practice, practice, and try to form some more outgoing habits. Sometimes all a shy person needs is a "partner in crime" - someone to diffuse the awkwardness, or similar. Also, for some reason, an activity far from home can be more appealing than one right up the street.
These activities will help by setting the thoughts and memories of these good experiences against the negative thoughts your friend usually has about situations like that. The more good experiences they have, the less impact the prior contributing factors will have; this is basically what CBT does, except that it tries to make you more conscious of the negative thoughts you have so that you can purposely counteract them. So your activities could have elements of CBT without you even trying.
Patience is key; your friend has most likely been living with this all her life, so it will probably take some time for her to come out of her shell. And sometimes shy people just don't. You will at least have gotten to spend some quality time with your friend.