It will take a while.
It could go into orbit at ca 0.5 light year distance, you won't see relevant effects for millions of years.. but you don't want that for your story (=dull), suppose worse case, your neutron star will travel straight into the sun, in the plane of the ecliptica (where planets are)
Neutron stars generally don't travel at near light speed. That could be different (you may decide for your story), but this is a solar mass object, it needs very bizarre circumstances to travel really fast through space. For below answer, I assume 1-2% c max speed.
Discovery: we'll see it coming in from light years away
There won't be gravitational waves, unless it's a twin (binary) and it certainly will be a big deal.. when it comes within 3ly it will disturb the solar system's debris region known as the Oort cloud, and that would be detected by astronomers. Rogue comets could collide on it, or pass it at high speed, in arbitrary direction.
Time scope: depending on approach velocity, 10 years
Academic subject at first..
Beyond one light year, that is 63000AU (sun-earth distances), there won't be relevant effects. It depends on its mass, when it starts to become a practical issue for Earth's inhabitants. To bridge this distance to the Solar system, it could take your Neutron star many years to reach the solar system, depending on the speed you set in your story.
Within ca 400AU, say 10x Pluto distance, a solar mass object will start disturbing the outer planetary orbits. I would become very concerned when that happens. It has come too close.
Time scope changes are noticed: years.
First effect noticed: climate and seasons.. but it depends on speed
When your neutron star comes within 80AE or closer, Earth's orbit will become distorted, with uncertain outcome. Earth could enter an elliptical orbit. That would have severe effects on climate. But again, it depends on approach speed of the neutron star, wether Earth will see the new winter or new summer (>1 year)
The Kuiper Belt, another debris zone, will be heavily disturbed when the neutron star is approaching Pluto distance. The density of the neutron star will cause matter to get high speed orbits and slinged into the solar system, Earth and moon will see more frequent meteor impacts and high speed impacts, which can have a devastating effect.
Time scope changes are noticed: months.
Apocalyptic consequences when it comes too near
Climate will not be your only concern. At a distance of less than 10AE, its radiation could start playing a devastating role now, destroying the ozon layer of Earth and baking every animal in UV radiation.
At say, Jupiter distance, a second solar mass object will affect the sun, causing matter to spiral in from the sun. The inner planets will be caught into orbit, or collide onto the neutron star. There will be a dangerous accretion disk, multiple deadly impacts.. and the Solar system will completely merge into the neutron star.
Time scope changes are noticed: WEEKS.