As others have pointed out, astronomical observations are the key.
Astronomers who went through the portal would first look for the familiar constellations seen from Earth, to eliminate the possibility that the other side of the portal is on Earth. A They would also search for planets, to see if the planet was in a solar system similar to Earth's, or one quite different. I think that even small telescopes should enable the identities of planets with those in our solar system to be confirmed or disproved.
The next step, or on taken by another team of astronomers, would be to scan the skies for dim patches of light. The large and small Magellanic Clouds and the Andromeda Galaxy should be visible to the naked eye from anywhere in the Milky way galaxy, except where hidden by the galactic core.
If they spot them, they will examine them with telescope to confirm their identity, and they will search with telescopes for other galaxies which are conspicuous from Earth.
They can search for globular star clusters, seeking to identify any that they find with the globular clusters surrounding our Milky Way Galaxy. If they can identify two or three of the globular clusters of the Milky Way Galaxy, measuring the angles between them should enable them to calculate the position of the planet relative to the Earth.
On Earth, the Sun is the strongest source of almost all frequencies of electromagnetic radiation, because it is so much closer than other stars and extra solar astronomical objects. On any other Earth like planet, the star or stars in the system will be much stronger in most radiation bands than any more distant objects.
Among the brightest objects in radio frequencies seen from Earth are the galaxies Centaurus A (NGC 5128) about 10-16 million light years from Earth, and Virgo A (M87 and NGC 4486) about 54 million light years from Earth. They are also relatively prominent in visual light, making it easier to identify the radio sources with known objects. The angle between Centaurus A and M87 can be used to calculate the position of the planet relative to Earth.
Pulsars can also be used to find the planet's location.
Pulsar maps have been included on the two Pioneer plaques as well as the Voyager Golden Record. They show the position of the Sun, relative to 14 pulsars, which are identified by the unique timing of their electromagnetic pulses, so that our position both in space and in time can be calculated by potential extraterrestrial intelligences. Because pulsars are emitting very regular pulses of radio waves, its radio transmissions do not require daily corrections. Moreover, pulsar positioning could create a spacecraft navigation system independently, or be used in conjunction with satellite navigation.
X-ray pulsar-based navigation and timing (XNAV) or simply pulsar navigation is a navigation technique whereby the periodic X-ray signals emitted from pulsars are used to determine the location of a vehicle, such as a spacecraft in deep space. A vehicle using XNAV would compare received X-ray signals with a database of known pulsar frequencies and locations. Similar to GPS, this comparison would allow the vehicle to calculate its position accurately (±5 km). The advantage of using X-ray signals over radio waves is that X-ray telescopes can be made smaller and lighter.1 Experimental demonstrations have been reported in 2018.
If Pulsars can be used to find the positions of space craft within our solar system, they can be used to find the positions of planets orbiting distant stars. But the atmosphere of a habitable planet would stop most X-rays from reaching the surface, so the astronomers couldn't be able to study Pulsar X-rays from the surface but would have to put detectors in orbit, which would be very difficult with what they could move through the portal.
The small size of the radio telescopes they could take through the portal would mean they would have to concentrate on the brightest radio sources. Both Centaurus A and Virgo A were discovered by 1950 with small radio telescopes, and so both are very bright radio sources, and far enough away to be similarly bright everywhere in our galaxy, and close enough to have large differences in angle as seen from different parts of our galaxy.