Can you buy train tickets online for the Copenhagen to Malmö route?

The vast majority of trains are run by Øresundståg. This line runs Malmö - Kastrup (CPH Airport) - Central Copenhagen (Kobenhavn H). In the direction Malmö-Copenhagen these are running normally: Malmö stopping at the airport and on to central Copenhagen. In the direction Copenhagen-Malmö, the Swedish Government's implementation of border controls means all passengers have to go through checks at Kastrup Airport where railway staff are checking all passengers ID's before crossing the border to Malmö, all train tickets are valid in local public transportation (bus, metro, other trains) from central Copenhagen to the airport you can use one of those methods to get to Kastrup Airport or take an Øresundståg train, though you'll have to get off at Kastrup Airport go through the check and get back on a different Øresundståg train.

There are a few rare SJ X2000 trains that also connect the two cities, these are a bit more expensive (from around 150 SEK one-way) and they're not really faster. The immigration check on these happens in Copenhagen in the Copenhagen-Malmö direction so at the moment they have the advantage of being the only direct trains in from central Copenhagen in the Copenhagen-Malmö direction.

XJ2000 trains prices may vary by day and how early you book, the Øresundståg has flat pricing between the cities. You can buy tickets for both the Øresundståg and XJ2000 trains from SJ online or at vending machines in either city. Swedish local public transit fares (Skånetrafiken's zone-based system) are also valid on the Øresundståg trains, these are only cheaper if you pay by getting a Jojo Card in Malmö (one-way fare is 100 SEK, but getting the card is kind of a hassle to figure out) or if you're party a party of two or more in which case you can use the Duo discount ticket, bringing the oneway down to around 100 SEK each (or 90 SEK each for Duo using a Jojo card). Skånetrafiken tickets also cover connecting to and from local public transit within the same zones on both sides of the Sound.

Since you are seem interested in saving money, it might also be worth checking out the handful of bus companies connecting the cities. The main draw back of these is they are less frequent than trains.

• Gråhund Bus 100 DKK for a day return ticket Malmo-Copenhagen-Malmo. Pay cash on departure. Seems they aren't running on Sundays currently.
• Swebus generally around 100 SEK one way.
• Nettbus generally around 90 SEK one way.
• FlixBus generally around €5 one way, only two buses and day and may be more prone to delays as the connection is part of a longer line.

I just tried working my way through both dsb.dk (Danish railways) and sj.se (Swedish railways). The Swedish site works like a charm as it did for me many times; you can select your train, buy a ticket, pay by credit card etc. The price on a random day in November for me was SEK 111, which amounts to €11.52 as of today’s exchange rate and Google. Prices in SEK shouldn’t fluctuate much. (For the journey, you should really go for the Øresundståg.) Since I was unable to get to anything that seemed like buy ticket on the Danish site, I would recommend using the Swedish one.

The Øresundståg call at Ørestad, Tårnby and Kastrup (the airport) on the Danish side of the bridge from Copenhagen central and in Hyllie and Triangeln on the Swedish side before Malmö. While it may be beneficial to board at a different station because that is where your hotel is, the price seems to be the same. Long-distance trains that arrive from or continue to Stockholm definitely call in Copenhagen and Malmö; before the reimplemented ID checks they also called in Kastrup. Even though the Swedish government has since relaxed its ID check requirements, I have not checked if X2000 trains call in Kastrup again. But then again, it shouldn’t be worth it to take an X2000 on that short a route.

Note on the border controls: I travelled Kastrup–Malmö by Øresundståg in July 2019 and was pleased to see that the Swedish requirements had been relaxed. This means: while the train company was previously required to check IDs of travellers before they boarded the train at Kastrup, this is no longer the case and trains travel normally on the Copenhagen–Malmö route.

Furthermore, Swedish police now perform far fewer ID checks on passengers arriving at Hyllie (the first station in Sweden). While a year ago most trains were checked my impression this summer was that most trains were ignored. (I stayed near Hyllie station and used it/spotted trains there a couple of times; only once was police there to perform an inspection.)

I believe you can use Rejsekort in trains between Malmö and Copenhagen.

http://rejsekort.dk/

But that's just for convenience of course, but also for the public transport in Copenhagen it's cheaper and more convenient. You should be able to get the anonymous without problems.