Does a blueshift mean that time goes faster? This is a follow-up question to this answer.
The assumption in this answer is that time dilation always causes a small redshift when an observer looks at an object moving at a significant fraction of the speed of light when not taking into account the shifts caused by the directional Doppler effect.
So, if time going slower always causes a redshift, does that mean that if we see a blueshift it means that time appears to move faster?
In other words, if B, that is far away from A, moves towards A really fast, A will appear to be blueshifted to B due to the relativistic doppler effect and thus B will see A's time moving faster?
The confusion I have is linking the concepts of redshifts and blueshifts with time going slower and faster.
 A: 
So, if time going slower always causes a redshift, does that mean that if we see a blueshift it means that time appears to move faster?

Yes.
The machine that produces the wave-crests that appear to follow each other extra rapidly, appears to work extra rapidly.
For example if wave-crests appear to follow each other at one nano-seconds intervals, then the machine that produces those wave-crests appears to produce one wave-crest each nano-second. I mean, when looked very closely through a big telescope, then the machine can be seen to do that.
Those machines that appear to person X to work extra rapidly work extra slowly according to person X. I mean, person X subtracts the directional blue shift, and notes that there is a redshift.
A: The Doppler effect relates to the rate at which events appear to be happening when viewed from some distance, rather than the rate at which they are happening, and the difference between the two rates is caused by the time it takes for the information about what is happening to travel from the source to the observer.
If you are moving towards a planet and you were able to study a clock on it through a powerful telescope as you approached, the clock would appear to be running fast. If you turned round and started moving away from the planet, the clock would seem to be running slow. Clearly the real tick-rate of the clock does not change.
To see why it works this way, suppose I lived a long way from you, and wrote you a letter every day, the letters each taking five days to arrive by post. After an initial delay of five days you will receive my first letter, and after that the letters will arrive daily, as each takes the same time to travel to you. If, however, I started travelling towards you, so that each day that I travel knocks a day off the postage time, then at some point you will receive five letters from me all on the same day. Although I wrote them a day apart, each successive letter has spent a day less in transit, so they all arrive at once. To you, the rate at which I have been writing letters seems to have speeded up.
Conversely, if I had gone travelling in the opposite direction, so each day I added one day more to the postage time, my letters would start arriving with you every two days, even though I was writing them daily, because each successive letter spends a day longer in transit. My rate of latter writing would appear to have slowed down.
My rate of writing letters is always one per day, but the rate at which you receive them is speeded up if I am moving towards you or slowed down if I am moving away.
A: This spacetime diagram may help.
The diamonds represent the light signals in a light-clock.
The resulting diamonds give us a graphically accurate visualization of the ticks of time and space according an observer. The rotated graph paper helps us draw these diamonds for different observers.
I use =(3/5) to make the arithmetic easier.

It was somehow arranged that
two astronauts will pass each other inertially
so that they meet briefly when they turn 30.
They broadcast their clock readings on their birthdays before and after the meeting.

What does each astronaut see (visually), based on the transmitted signals from the distant clock and the local clock?
Before they meet,
each astronaut receives two transmissions for every one year.
This is akin to blueshift... a shorter period between receptions of birthday-images... a higher frequency of receptions.
Specifically:
Before they met

*

*at age 26, the image from the local clock is 26 but image from the distant clock is 22.


*at age 27, the image from the local clock is 27 but image from the distant clock is 24. (The distant clock's image of 23 was seen at age 26.5.) ... and so on.
When they meet,

*

*each astronaut receives "30" from the local clock and the distant clock.

After they met,

*

*at age 34, the image from the distant clock is 32

*at age 35, the image from the distant clock is 32.5

*at age 36, the image from the distant clock is 33 ... and so on.

After they met,
each astronaut now receives one transmission for every two years.
This is akin to redshift... a longer period between receptions of birthday-images... a lower frequency of receptions.
