Approximately how much does a young dachshund cost a month?

The initial puppy vaccinations, de-worming, micro-chipping, neuter/spay are what cost way more than any "monthly" maintenance of a healthy dachshund.

After that, it is good dog food, not supermarket junk, a name brand flea/tick preventative (Frontline Plus or Advantix, about 8.- per month) and heartworm preventative (generic is fine, about 6.- per month).

You also need to have a reserve fund in case your dog gets sick or hurt and needs vetting.

If you can find one on Petfinder.com or from a dachshunds rescue, the neuter/spay and vaccinations, as well as micro-chipping will likely already be done. Dachshunds live a long time if well cared for and healthy, so a grown dachshunds is fine, doesn't have to be a puppy. Lots less trouble and work too, look up dachshunds on Petfinder and see some really cute ones.

Just make sure you can financially provide for it and give it a good Forever Home.

Food-wise depending on the food you buy $30 should last a couple of months for a mid-range food.

Vet-wise it can be anywhere from cheap to astronomical. My previous one no health issues that required a vet (his back did go out but crate rest cured him). If yours has a disk rupture it could be upwards of $5000, and I think that can be where you need the surgery to block pain or you can skip the surgery and have a wiener on wheels. If you end up with a paralyzed Dachshund you can expect a lot of UTIs and diaper costs. If your vet works with you on the UTIs while you learn to effectively empty the bladder you can get away comparatively cheap, if they want to soak you for testing every time you have a UTI you can go broke (it's a good time to be assertive or find a new vet if they try to soak you for a special needs dog).