IST topography

I have stayed in the Marmora Guesthouse. It is delightful, but not suited to someone with mobility / staircase difficulties. The stairs are steep and the breakfast deck [which is beuatiful] is up 3 [or4] floors. No elevator. Also, that area has mostly cobblestone road surfaces.
Topkapi is not the high point in Sultanamet area. That would be the Grand Bazaar. I am puzzled by your preference for going downhill because when you return to the hotel it will be uphill, or vice versa. Perhaps I do not understand what you meant to convey.
On your map, locate the Underground Cistern. Think of that as 'the corner'. That is where the modern streetcar turns from Divan Yolu to Alamdar [street names change with distance, and spelling, be careful about them]. Alamdar is the street that goes down to the waterfront [changing names] where there are many ferries, then the Spice Market, and the Galata Bridge across the Golden Horn to Karakoy. Yolu goes towards Grand Bazaar and University and many miles beyond.
From that corner it is essentially flat walking to Hagia Sophia across the street and less than 500 meters, through park to Blue Mosque / Hippodrome / Turkish Islamic Arts Museum or the other direction past historical buildings, to Topkapi or Archaeological Museum. Purchase Museum Passes, it is easy to spend 2-4 days in this small, easily walked area.
Look for a hotel close to that corner, there are many and some have elevators. Having elevators often comes with other amenities that result in prices being higher than a Guesthouse. Communicate with the hotel to ensure that the elevator is functioning and functional for your needs. If noise is an issue, be careful about rooms that face onto main streets. Close to that corner, on Yolu, there are many restaurants, money exchanges, tourist companies etc.
From that corner towards Grand Bazaar [on Yolu] it is slightly uphill. Sidewalks are often broken or require a curb step. The streetcar is very efficient if your friend can manage two steps. Taxis are cheap when two people are using it. Istanbul is not wheelchair friendly, but often one-three steps will make places accessible.
Read my postings under "Istanbul 2-3 days"
Buy RS Istanbul guidebook and read it twice before you go. It is full of useful information, will improve your visit, AND save you money! What a wonderful combination!
Have a wonderful visit.
Harvey

Galata, Beyoglu, Karakoy is hilly with some very steep segments of road. Istiklal, a main street which is pedestrian only, proceeds from Taksim Square for several Kms. It is basically flat. Enjoyable to see the people of Istanbul out for a stroll. Enjoyable architecture of the 1800's. It has a delightful historical tram running own the middle. It ends close, but not flat, to Galata Tower.
These areas have some active nightlife and entertainment and some visitors enjoy it. I prefer Sultanahmet.

Hey Harvey,

Thanks for sharing your knowledge of the city, I also read your post, 2-3 ds in IST. Looks you know it very well. Nice to hear that Istictal's mostly flat. Am convinced to just stay both times at Sultanahmet. I just wanted my friend to go to major cites downhill, back to a hotel uphill, not vice versa. Grateful for the hint that elevator may be out of order.... It's unpredictable, though....Will book w/free cancellation & call hotel close to the last date of cancellation, I guess. My experience's that big city noise's inescapable.... I traveled through Greece for 21 days, and street noise was a huge problem just everywhere. Not so much cars, as motorcycles....This buzzz penetrated just throughout anything, from ANY distances...

If you are going in warm weather, chances are you'll have the windows closed and AC on in your room, so noise shouldn't be a big issue. Sultanahmet is pretty quiet during the night, unless you are over pub that's open till all hours or facing the tram line that starts early in the morning. There are some inclines in Sultanahmet but the only steepish one I remember is from the Arista Bazaar (small but quality) up to the main street/tram line. Topkapi is mostly level, but a lot of walking. I remember going up and down flights of stairs at the on-site cafe (great views) both to tables and to the toilet.

Be watchful with taxi drivers. Some are dishonest and will try to scam you. For instance if you are paying with a large note, tell the guy "here's a 50."

Hi Ann, I visited YouTube and searched for "Istanbul 2 days" and spent a delightful hour watching a selection of videos produced by various travelers. They brought back wonderful memories and increased my desire to return to Turkey and Greece [the Aegean world]. They are a wonderful source of information, especially visual information such as "How flat is it?" "What do the streetcars look like?" "What does delicious Turkish food or sweets look like?".
One video showed/mentioned the Seven Hills Restaurant and the beautiful view from the roof top restaurant, almost 360 degrees of Blue Mosque - Hagia Sophia - Bosphorus waterway. It is more expensive, but just consider it a surcharge for visual beauty; perhaps go for coffee and desert at sunset. I think it has an elevator, and maybe, I am not sure, there might be a hotel on the lower floors. It is across the street from The Four Seasons Hotel [which is a converted building and does not look like a hotel] which might be more well known. It is within 2-3 short blocks of The Marmora Guesthouse which you had mentioned.
Harvey

Hi Ann, you inspired me to dig out my copy of RS Istanbul. I reread most of the "Sleeping" chapter and remembered what a great resource of information it is, not just individual hotels but what is typical, what is assumed, what is negotiable. Such as, many hotels offer free transportation from the airport, however remember the new airport is much further so this may have changed. Many reduce rates for longer stays, direct booking, paying cash not credit card, mentioning RS guidebook, etc.
My copy was published in 2012 so info about specific hotels may have changed.
One hotel that RS thought well of and which might suit your needs [it has a elevator and an excellent view from the rooftop deck] is Adamar Hotel, Yerebatan Cadd 47, which is located close to "the corner" I mentioned and is not on a main road. If you search on one of the hotel.com or similar site you will be able to see multiple hotels in that few blocks area.
Harvey

Hey Harvey, Chani
Yep, I am digging into “RS” book right now. Not quite a RS, as it’s written by IST guides, Arans. Though edited by another EU guide to fit for USA travelers and all the venues’ve checked out by RS himself...Mine is 7 th Ed, 2017. Yep, some hotels listed mentioned that elevators present: hotel stone, lausos. Adamar is mentioned, too...a bit pricey though, but it seems to be quiet location. Will dilute reading w/ tube viewing, visual helps ;-)

Thanks Chani f reminding about taxi scams, Arans mention this conundrum, too...

We stayed at the Sirkeci Konak hotel in 2013 and absolutely loved it. Had elevator, wonderful breakfasts, airport shuttle. It's in the heart of Sultanahmet, easy walking to all the major sights (no major hills) and close to the tram station. Almost next door to Topkapi Palace. Wonderful staff. They offered free Turkish cooking classes, plus complimentary tea and snacks at tea time. I'd gladly stay here again.