I am in no way an expert for NTFS, but I'd like to give some general guidelines.
1) I have seen a lot of things regarding PC software, but never heard of an application which could extend NTFS partitions to the beginning of a drive without moving the data.
2) Whether such application does exist or not: You absolutely should not do such sorts of things without taking a full backup of the disk before.
Extending, shrinking or moving partitions is always a high-risk operation, so it is strongly recommended to take a backup of the whole disk before doing such things, and you should keep the time the operation takes as short as possible.
So, even if there was such an application which could do what you asked about, it still would have to (recursively) adjust a great bunch of pointers, and that would take its time, and therefore I would not use it.
In your case, I would do the following:
Minimum (bad) solution
a) Delete partition A
b) Re-create partition A with a size which leaves enough room for partition B (after that step, there will be some room between partition A and partition B)
c) Copy all data from partition B to partition A (this could fail if partition A is too small now; in this case, that solution actually isn't one)
c) Delete partition B
d) Re-create partition B with bigger size (after that step, there will be no room between partition A and partition B)
e) Copy all data from partition A to partition B
a) Take a full backup at least of partition B
b) Delete both partitions
c) Re-create both partitions with sizes as needed
d) Restore the backup of old partition B to new partition B
a) Buy a new disk
b) Partition the new disk, thereby creating partitions with sizes as needed
c) Take a full backup of partition B of the old disk
d) Restore that backup into partition B of the new disk
e) Keep the old disk in a safe place in case something went wrong. After having worked with the new disk for a month or so, when knowing that everything is well, you could use the old disk for other things.
So why is the first solution bad?
First, as you may know, NTFS is quite complex. There are things like ACLs, hard links, symbolic links, shortcuts, sparse files and alternate streams, to name a few of them. Heck, even the integrated Windows Explorer shows wrong directory sizes if there are hard links! Personally, I am knowing exactly two programs which reliably copy data without losing such information; both are command line, and their usage is fairly difficult to understand. I definitely would only recommend them for advanced users.
Second, if your partition B is where your OS resides, just copying its data (even with one of the tools I mentioned) to partition A won't make the OS boot from partition A.
But even that bad solution will cut the time the high-risk operations (i.e. resizing the partition) will take to a reasonable amount.
Some additional notes:
These are my personal opinions and best-practices I have developed over the years (I have done such things more than one time)
When saying "full backup", I of course mean image-type backups (as opposed to file-based backups)
There are great free and commercial tools for taking image-based backups. For example, look for dd, CloneZilla or TrueImage.
When restoring full backups, depending on the software you use, the destination partition might need to have at least the size the original partition had.
After having restored a full backup to a partition, if that partition is greater in size than the partition the backup came from, you eventually (as a second step) have to extend the file system to make it use the whole partition (yes, having a file system which is limited to 100 GB on a partition which is 200 GB in size is indeed possible); whether this is necessary or not depends on the backup software used.
And finally, once again, think about it: You could run backups overnight, so what is the problem with the time that takes compared to losing valuable personal data (e.g. the photos which you have taken when you have been on Mount Everest's peak)? And if you don't have the money for a new disk, then you could ask a friend if he will let you use his external disk for a day or two ...