The place to check is vedur.is, the website of Iceland's Met Office. You will find the weather forecast and observations (including aurora) as well as earthquake observations etc.
Accommodation is generally expensive in Greater Reykjavik and Iceland in general and not necessarily of the best quality for what you pay. It is something that you simply have to put up with. Use booking.com or airbnb, read reviews, and choose the location carefully.
Public buses are infrequent and expensive. So even if your hotel is not very far, if it is not walking distance from the university, you may be in mild trouble. (Notice that you need Reykjavik University, not the University of Iceland; both are located next to the domestic airport, but at opposite ends).
In general, any location in Reykjavik districts 101, 102 is fine and 105 and 107 too. (Residents of Greater Reykjavik tend to identify city districts by postal codes.) Walking from the center of 101 to the university takes about 35 mins.
Hotel Natura is between the city center and the university, also very close to the terminal of Reykjavik Excursions (Icelandia) at BSÍ.
No-one uses cash in Iceland. We recommend that you do not exchange currency or withdraw Icelandic cash from an ATM.
The Keflavík international airport (KEF) is next to the town of Reykjanesbær some 50 kms from Reykjavik.
You get to the city with the Flybus service of the commercial operator Reykjavik Excursions (Icelandia). The transfer costs about 4000 ISK one way. If you add a connection to your hotel/guesthouse, that is another 1000 ISK. There is a ticket booth in the arrivals hall and buses depart close to the entrance. There is no need to book the ticket in advance on the way to Reykjavik. (On the way back to the airport you might want an advance booking, including a pickup from your accommodation.)
There is also the line 55 of the Icelandic public bus system Strætó. The ride costs 2680 ISK one way. The bus stop is a bit hidden close to the airport hotel Aurora. This bus is infrequent, but you can check the schedules. You can buy the ticket from the driver on the vehicle and it is also good for changing in Reykjavik.
Unless you are Norwegian, you do not want to think of taking a taxi from/to the airport if you are not 3 people sharing.
The center is small. For places a bit further, you may need to use city buses of Strætó. A single ticket costs 670 ISK (335 ISK for people aged 67 or older), it is valid for changes in the Greater Reykjavik area during 75 mins. You can use the Klappið app or pay contactless on the bus (only the full fare 670 ISK). Drivers do not sell tickets.
From the center to the university you can get with bus 8 from Gamla Hringbraut to HR. This is just 2 stops, but it is a walk of 20 mins along the perimeter of the domestic airport. To Gamla Hringbraut, you have to walk or take some other bus (1, 3, 5, 6, 15), depending on where you are staying.
In evenings and on weekends, one gets to/from HR with bus 5 instead. This route starts in a far-out suburb in the East of Reyjavik, stops at Ártún (a major stop for changing between lines), passes through Laugardalur, the city center (the Laugavegur stop) and Gamla Hringbraut.
Eating out is expensive, but nicer restaurants are nice, no doubt.
The useful foodstore chains are Bónus, Krónan and Nettó. Opening times vary, some stores open later and close earlier than you would expect. At 10-11 and Krambúð stores (open long or round the clock) you will find a limited selection and high prices.
The best and also the cheapest fun in Reykjavik are the municipal geothermally heated swimming pools. The main ones in the city are Laugardalslaug, Sundhöll Reykjavíkur and Vesturbæjarlaug. You get an outdoor swimming pool (25 m in most places, 50 m in Laugardalslaug; the water temperature is 26 C all year round), hotpots (38-42 C) and steam saunas. Sundhöll Reykjavíkur is the most central one. Laugardalslaug is the biggest and most fun. A single ticket costs 1380 ISK.
All swimming pools are open from early morning until 22.00 and this means you can stay in the pool until 22.00 and then have 15 extra minutes to change and rush out.
Just behind Reykjavik University, there is the Nauthólsvík geothermal beach. Here the fun consists in a hotpot and dips in the ocean (seawater ~5 C), there is also a steam sauna. This place is open 11-19 daily, until 16 on Sat, closed Sun and Mon. A single ticket is 920 ISK (free for people aged 67 years or older).
There is also the Blue Lagoon if you want to pay a lot more. It is closer to the airport than to Greater Reykjavik and you will need a whole morning or afternoon to go there. The Sky Lagoon is a fancy spa in Kópavogur.
If you are into this type of thing, take your swimming gear with you!
There is lots to see both in Reykjavik and out in the countryside. See Visit Reykjavik and Visit Iceland. Many operators (Reykjavik Excursions, Gray Line, Bus Travel, Arctic Adventures and others) offer day tours to the countryside. The most famous such tours are the Golden Circle (Þingvellir national park, Geysir, Gullfoss waterfall) and the Southern Shore Adventure (Seljalandsfoss, Skógafoss waterfalls, Sólheimajökull glacier tongue, Reynisfjara black sand beach). Alternatively you can rent a car.