Here's what I would do, but it depends on what else you have available or would be willing to purchase.
Assuming that you're going with a new 1TB SSD (because that's what I would do), I would also buy a 2TB (or higher) to use as the Time Machine drive. I guess if you go with 480GB, the 1 TB external that you have might be fine.
If you're not interested in a cloning your 128GB drive onto your new SSD, just copy over all of your important files from the SSD to the 1 TB. Install the new SSD and put whatever macOS you want. Redownload all of your apps and settings to set the new drive the way you want. Copy over the files from the 1TB onto the new SSD.
Re-format the 1TB and use as a Time Machine. Make sure that you have everything you need off of the drive and that your new SSD is running fine before doing this.
If you want to clone your 128GB drive, you'll need either another hard drive or an enclosure for the SSD. This is easier than the above, but will require extra hardware. Cloning software can be used on a trial period, but it's handy to have full access. I use SuperDuper, but CarbonCopyCloner is also popular.